diff --git a/pkg/expression/column.go b/pkg/expression/column.go
index 68f3eb900d416..b9d883bc61388 100644
--- a/pkg/expression/column.go
+++ b/pkg/expression/column.go
@@ -36,6 +36,7 @@ import (
 
 var (
 	_ base.HashEquals = &Column{}
+	_ base.HashEquals = &CorrelatedColumn{}
 )
 
 // CorrelatedColumn stands for a column in a correlated sub query.
@@ -246,6 +247,31 @@ func (col *CorrelatedColumn) RemapColumn(m map[int64]*Column) (Expression, error
 	}, nil
 }
 
+// Hash64 implements HashEquals.<0th> interface.
+func (col *CorrelatedColumn) Hash64(h base.Hasher) {
+	// correlatedColumn flag here is used to distinguish correlatedColumn and Column.
+	h.HashByte(correlatedColumn)
+	col.Column.Hash64(h)
+	// since col.Datum is filled in the runtime, we can't use it to calculate hash now, correlatedColumn flag + column is enough.
+}
+
+// Equals implements HashEquals.<1st> interface.
+func (col *CorrelatedColumn) Equals(other any) bool {
+	if other == nil {
+		return false
+	}
+	var col2 *CorrelatedColumn
+	switch x := other.(type) {
+	case CorrelatedColumn:
+		col2 = &x
+	case *CorrelatedColumn:
+		col2 = x
+	default:
+		return false
+	}
+	return col.Column.Equals(&col2.Column)
+}
+
 // Column represents a column.
 type Column struct {
 	RetType *types.FieldType `plan-cache-clone:"shallow"`
@@ -458,11 +484,11 @@ func (col *Column) Hash64(h base.Hasher) {
 	h.HashInt64(col.ID)
 	h.HashInt64(col.UniqueID)
 	h.HashInt(col.Index)
-	if col.VirtualExpr != nil {
+	if col.VirtualExpr == nil {
 		h.HashByte(base.NilFlag)
 	} else {
 		h.HashByte(base.NotNilFlag)
-		//col.VirtualExpr.Hash64(h)
+		col.VirtualExpr.Hash64(h)
 	}
 	h.HashString(col.OrigName)
 	h.HashBool(col.IsHidden)
@@ -488,12 +514,12 @@ func (col *Column) Equals(other any) bool {
 	}
 	// when step into here, we could ensure that col1.RetType and col2.RetType are same type.
 	// and we should ensure col1.RetType and col2.RetType is not nil ourselves.
-	ftEqual := col.RetType == nil && col2.RetType == nil || col.RetType != nil && col2.RetType != nil && col.RetType.Equal(col2.RetType)
-	return ftEqual &&
+	ok := col.RetType == nil && col2.RetType == nil || col.RetType != nil && col2.RetType != nil && col.RetType.Equal(col2.RetType)
+	ok = ok && (col.VirtualExpr == nil && col2.VirtualExpr == nil || col.VirtualExpr != nil && col2.VirtualExpr != nil && col.VirtualExpr.Equals(col2.VirtualExpr))
+	return ok &&
 		col.ID == col2.ID &&
 		col.UniqueID == col2.UniqueID &&
 		col.Index == col2.Index &&
-		//col.VirtualExpr.Equals(col2.VirtualExpr) &&
 		col.OrigName == col2.OrigName &&
 		col.IsHidden == col2.IsHidden &&
 		col.IsPrefix == col2.IsPrefix &&
diff --git a/pkg/expression/column_test.go b/pkg/expression/column_test.go
index 78dcd8c061443..153cd6f929b43 100644
--- a/pkg/expression/column_test.go
+++ b/pkg/expression/column_test.go
@@ -415,8 +415,7 @@ func TestColumnHashEquals(t *testing.T) {
 	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
 	require.False(t, col1.Equals(col2))
 
-	// diff VirtualExpr
-	// TODO: add HashEquals for VirtualExpr
+	// diff VirtualExpr see TestColumnHashEuqals4VirtualExpr
 
 	// diff OrigName
 	col2.Index = col1.Index
@@ -468,3 +467,29 @@ func TestColumnHashEquals(t *testing.T) {
 	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
 	require.False(t, col1.Equals(col2))
 }
+
+func TestColumnHashEuqals4VirtualExpr(t *testing.T) {
+	col1 := &Column{UniqueID: 1, VirtualExpr: NewZero()}
+	col2 := &Column{UniqueID: 1, VirtualExpr: nil}
+	hasher1 := base.NewHashEqualer()
+	hasher2 := base.NewHashEqualer()
+	col1.Hash64(hasher1)
+	col2.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, col1.Equals(col2))
+
+	col2.VirtualExpr = NewZero()
+	hasher2.Reset()
+	col2.Hash64(hasher2)
+	require.Equal(t, hasher1.Sum64(), hasher2.Sum64())
+	require.True(t, col1.Equals(col2))
+
+	col1.VirtualExpr = nil
+	col2.VirtualExpr = nil
+	hasher1.Reset()
+	hasher2.Reset()
+	col1.Hash64(hasher1)
+	col2.Hash64(hasher2)
+	require.Equal(t, hasher1.Sum64(), hasher2.Sum64())
+	require.True(t, col1.Equals(col2))
+}
diff --git a/pkg/expression/constant.go b/pkg/expression/constant.go
index 210b55186eff8..7b2599247f9c2 100644
--- a/pkg/expression/constant.go
+++ b/pkg/expression/constant.go
@@ -20,6 +20,7 @@ import (
 
 	perrors "github.com/pingcap/errors"
 	"github.com/pingcap/tidb/pkg/parser/mysql"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
 	"github.com/pingcap/tidb/pkg/util/codec"
@@ -29,6 +30,8 @@ import (
 	"go.uber.org/zap"
 )
 
+var _ base.HashEquals = &Constant{}
+
 // NewOne stands for a number 1.
 func NewOne() *Constant {
 	retT := types.NewFieldType(mysql.TypeTiny)
@@ -502,6 +505,50 @@ func (c *Constant) CanonicalHashCode() []byte {
 	return c.getHashCode(true)
 }
 
+// Hash64 implements HashEquals.<0th> interface.
+func (c *Constant) Hash64(h base.Hasher) {
+	if c.RetType == nil {
+		h.HashByte(base.NilFlag)
+	} else {
+		h.HashByte(base.NotNilFlag)
+		c.RetType.Hash64(h)
+	}
+	c.collationInfo.Hash64(h)
+	if c.DeferredExpr != nil {
+		c.DeferredExpr.Hash64(h)
+		return
+	}
+	if c.ParamMarker != nil {
+		h.HashByte(parameterFlag)
+		h.HashInt64(int64(c.ParamMarker.order))
+		return
+	}
+	intest.Assert(c.DeferredExpr == nil && c.ParamMarker == nil)
+	h.HashByte(constantFlag)
+	c.Value.Hash64(h)
+}
+
+// Equals implements HashEquals.<1st> interface.
+func (c *Constant) Equals(other any) bool {
+	if other == nil {
+		return false
+	}
+	var c2 *Constant
+	switch x := other.(type) {
+	case *Constant:
+		c2 = x
+	case Constant:
+		c2 = &x
+	default:
+		return false
+	}
+	ok := c.RetType == nil && c2.RetType == nil || c.RetType != nil && c2.RetType != nil && c.RetType.Equals(c2.RetType)
+	ok = ok && c.collationInfo.Equals(c2.collationInfo)
+	ok = ok && (c.DeferredExpr == nil && c2.DeferredExpr == nil || c.DeferredExpr != nil && c2.DeferredExpr != nil && c.DeferredExpr.Equals(c2.DeferredExpr))
+	ok = ok && (c.ParamMarker == nil && c2.ParamMarker == nil || c.ParamMarker != nil && c2.ParamMarker != nil && c.ParamMarker.order == c2.ParamMarker.order)
+	return ok && c.Value.Equals(c2.Value)
+}
+
 func (c *Constant) getHashCode(canonical bool) []byte {
 	if len(c.hashcode) > 0 {
 		return c.hashcode
diff --git a/pkg/expression/constant_test.go b/pkg/expression/constant_test.go
index 8be33601bae5d..b338d6238c1a2 100644
--- a/pkg/expression/constant_test.go
+++ b/pkg/expression/constant_test.go
@@ -25,6 +25,7 @@ import (
 	exprctx "github.com/pingcap/tidb/pkg/expression/context"
 	"github.com/pingcap/tidb/pkg/parser/ast"
 	"github.com/pingcap/tidb/pkg/parser/mysql"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
 	"github.com/pingcap/tidb/pkg/util/mock"
@@ -545,3 +546,30 @@ func TestSpecificConstant(t *testing.T) {
 	require.Equal(t, null.RetType.GetFlen(), 1)
 	require.Equal(t, null.RetType.GetDecimal(), 0)
 }
+
+func TestConstantHashEquals(t *testing.T) {
+	// Test for Hash64 interface
+	cst1 := &Constant{Value: types.NewIntDatum(2333), RetType: newIntFieldType()}
+	cst2 := &Constant{Value: types.NewIntDatum(2333), RetType: newIntFieldType()}
+	hasher1 := base.NewHashEqualer()
+	hasher2 := base.NewHashEqualer()
+	cst1.Hash64(hasher1)
+	cst2.Hash64(hasher2)
+	require.Equal(t, hasher1.Sum64(), hasher2.Sum64())
+	require.True(t, cst1.Equals(cst2))
+
+	// test cst2 datum changes.
+	cst2.Value = types.NewIntDatum(2334)
+	hasher2.Reset()
+	cst2.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, cst1.Equals(cst2))
+
+	// test cst2 type changes.
+	cst2.Value = types.NewIntDatum(2333)
+	cst2.RetType = newStringFieldType()
+	hasher2.Reset()
+	cst2.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, cst1.Equals(cst2))
+}
diff --git a/pkg/expression/expression.go b/pkg/expression/expression.go
index 07f83c1f06609..799dd58da850c 100644
--- a/pkg/expression/expression.go
+++ b/pkg/expression/expression.go
@@ -27,6 +27,7 @@ import (
 	"github.com/pingcap/tidb/pkg/parser/mysql"
 	"github.com/pingcap/tidb/pkg/parser/opcode"
 	"github.com/pingcap/tidb/pkg/parser/terror"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
 	"github.com/pingcap/tidb/pkg/util/generatedexpr"
@@ -42,6 +43,7 @@ const (
 	scalarFunctionFlag byte = 3
 	parameterFlag      byte = 4
 	ScalarSubQFlag     byte = 5
+	correlatedColumn   byte = 6
 )
 
 // EvalSimpleAst evaluates a simple ast expression directly.
@@ -170,6 +172,7 @@ const (
 type Expression interface {
 	VecExpr
 	CollationInfo
+	base.HashEquals
 
 	Traverse(TraverseAction) Expression
 
diff --git a/pkg/expression/scalar_function.go b/pkg/expression/scalar_function.go
index 3ec1c6cb7f8cd..f36581eabbcd1 100644
--- a/pkg/expression/scalar_function.go
+++ b/pkg/expression/scalar_function.go
@@ -26,6 +26,7 @@ import (
 	"github.com/pingcap/tidb/pkg/parser/model"
 	"github.com/pingcap/tidb/pkg/parser/mysql"
 	"github.com/pingcap/tidb/pkg/parser/terror"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/sessionctx/variable"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
@@ -35,6 +36,8 @@ import (
 	"github.com/pingcap/tidb/pkg/util/intest"
 )
 
+var _ base.HashEquals = &ScalarFunction{}
+
 // ScalarFunction is the function that returns a value.
 type ScalarFunction struct {
 	FuncName model.CIStr
@@ -673,6 +676,51 @@ func simpleCanonicalizedHashCode(sf *ScalarFunction) {
 	}
 }
 
+// Hash64 implements HashEquals.<0th> interface.
+func (sf *ScalarFunction) Hash64(h base.Hasher) {
+	h.HashByte(scalarFunctionFlag)
+	h.HashString(sf.FuncName.L)
+	if sf.RetType == nil {
+		h.HashByte(base.NilFlag)
+	} else {
+		h.HashByte(base.NotNilFlag)
+		sf.RetType.Hash64(h)
+	}
+	// hash the arg length to avoid hash collision.
+	h.HashInt(len(sf.GetArgs()))
+	for _, arg := range sf.GetArgs() {
+		arg.Hash64(h)
+	}
+}
+
+// Equals implements HashEquals.<1th> interface.
+func (sf *ScalarFunction) Equals(other any) bool {
+	if other == nil {
+		return false
+	}
+	var sf2 *ScalarFunction
+	switch x := other.(type) {
+	case *ScalarFunction:
+		sf2 = x
+	case ScalarFunction:
+		sf2 = &x
+	default:
+		return false
+	}
+	ok := sf.FuncName.L == sf2.FuncName.L
+	ok = ok && (sf.RetType == nil && sf2.RetType == nil || sf.RetType != nil && sf2.RetType != nil && sf.RetType.Equals(sf2.RetType))
+	if len(sf.GetArgs()) != len(sf2.GetArgs()) {
+		return false
+	}
+	for i, arg := range sf.GetArgs() {
+		ok = ok && arg.Equals(sf2.GetArgs()[i])
+		if !ok {
+			return false
+		}
+	}
+	return ok
+}
+
 // ReHashCode is used after we change the argument in place.
 func ReHashCode(sf *ScalarFunction) {
 	sf.hashcode = sf.hashcode[:0]
diff --git a/pkg/expression/scalar_function_test.go b/pkg/expression/scalar_function_test.go
index 810adbb3b569a..eb1920eacb6a2 100644
--- a/pkg/expression/scalar_function_test.go
+++ b/pkg/expression/scalar_function_test.go
@@ -19,6 +19,7 @@ import (
 
 	"github.com/pingcap/tidb/pkg/parser/ast"
 	"github.com/pingcap/tidb/pkg/parser/mysql"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
 	"github.com/pingcap/tidb/pkg/util/mock"
@@ -147,3 +148,40 @@ func TestScalarFuncs2Exprs(t *testing.T) {
 		require.True(t, exprs[i].Equal(ctx, funcs[i]))
 	}
 }
+
+func TestScalarFunctionHash64Equals(t *testing.T) {
+	a := &Column{
+		UniqueID: 1,
+		RetType:  types.NewFieldType(mysql.TypeDouble),
+	}
+	sf0, _ := newFunctionWithMockCtx(ast.LT, a, NewZero()).(*ScalarFunction)
+	sf1, _ := newFunctionWithMockCtx(ast.LT, a, NewZero()).(*ScalarFunction)
+	hasher1 := base.NewHashEqualer()
+	hasher2 := base.NewHashEqualer()
+	sf0.Hash64(hasher1)
+	sf1.Hash64(hasher2)
+	require.Equal(t, hasher1.Sum64(), hasher2.Sum64())
+	require.True(t, sf0.Equals(sf1))
+
+	// change the func name
+	sf2, _ := newFunctionWithMockCtx(ast.GT, a, NewZero()).(*ScalarFunction)
+	hasher2.Reset()
+	sf2.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, sf0.Equals(sf2))
+
+	// change the args
+	sf3, _ := newFunctionWithMockCtx(ast.LT, a, NewOne()).(*ScalarFunction)
+	hasher2.Reset()
+	sf3.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, sf0.Equals(sf3))
+
+	// change the ret type
+	sf4, _ := newFunctionWithMockCtx(ast.LT, a, NewZero()).(*ScalarFunction)
+	sf4.RetType = types.NewFieldType(mysql.TypeLong)
+	hasher2.Reset()
+	sf4.Hash64(hasher2)
+	require.NotEqual(t, hasher1.Sum64(), hasher2.Sum64())
+	require.False(t, sf0.Equals(sf4))
+}
diff --git a/pkg/expression/util_test.go b/pkg/expression/util_test.go
index f236e943a9d0b..1329149b60b3a 100644
--- a/pkg/expression/util_test.go
+++ b/pkg/expression/util_test.go
@@ -22,6 +22,7 @@ import (
 	"github.com/pingcap/tidb/pkg/parser/ast"
 	"github.com/pingcap/tidb/pkg/parser/model"
 	"github.com/pingcap/tidb/pkg/parser/mysql"
+	"github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/sessionctx/stmtctx"
 	"github.com/pingcap/tidb/pkg/types"
 	"github.com/pingcap/tidb/pkg/util/chunk"
@@ -661,3 +662,5 @@ func (m *MockExpr) MemoryUsage() (sum int64) {
 func (m *MockExpr) Traverse(action TraverseAction) Expression {
 	return action.Transform(m)
 }
+func (m *MockExpr) Hash64(_ base.Hasher) {}
+func (m *MockExpr) Equals(_ any) bool    { return false }
diff --git a/pkg/planner/core/BUILD.bazel b/pkg/planner/core/BUILD.bazel
index 6a7fa154f4ae5..b511fdfbde465 100644
--- a/pkg/planner/core/BUILD.bazel
+++ b/pkg/planner/core/BUILD.bazel
@@ -126,6 +126,7 @@ go_library(
         "//pkg/parser/terror",
         "//pkg/parser/types",
         "//pkg/planner/cardinality",
+        "//pkg/planner/cascades/base",
         "//pkg/planner/context",
         "//pkg/planner/core/base",
         "//pkg/planner/core/constraint",
diff --git a/pkg/planner/core/scalar_subq_expression.go b/pkg/planner/core/scalar_subq_expression.go
index 17dd07fdc1f1e..2a7f2bf16ecee 100644
--- a/pkg/planner/core/scalar_subq_expression.go
+++ b/pkg/planner/core/scalar_subq_expression.go
@@ -22,6 +22,7 @@ import (
 	"github.com/pingcap/errors"
 	"github.com/pingcap/tidb/pkg/expression"
 	"github.com/pingcap/tidb/pkg/infoschema"
+	base2 "github.com/pingcap/tidb/pkg/planner/cascades/base"
 	"github.com/pingcap/tidb/pkg/planner/core/base"
 	"github.com/pingcap/tidb/pkg/planner/core/operator/baseimpl"
 	"github.com/pingcap/tidb/pkg/types"
@@ -224,6 +225,29 @@ func (s *ScalarSubQueryExpr) ExplainNormalizedInfo() string {
 	return s.String()
 }
 
+// Hash64 implements the HashEquals.<0th> interface.
+func (s *ScalarSubQueryExpr) Hash64(h base2.Hasher) {
+	h.HashByte(expression.ScalarSubQFlag)
+	h.HashInt64(s.scalarSubqueryColID)
+}
+
+// Equals implements the HashEquals.<1st> interface.
+func (s *ScalarSubQueryExpr) Equals(other any) bool {
+	if other == nil {
+		return false
+	}
+	var s2 *ScalarSubQueryExpr
+	switch x := other.(type) {
+	case *ScalarSubQueryExpr:
+		s2 = x
+	case ScalarSubQueryExpr:
+		s2 = &x
+	default:
+		return false
+	}
+	return s.scalarSubqueryColID == s2.scalarSubqueryColID
+}
+
 // HashCode implements the Expression interface.
 func (s *ScalarSubQueryExpr) HashCode() []byte {
 	if len(s.hashcode) != 0 {