Quantization of Kleinian singularities can be realized as two different quantum Hamilto- nian reductions. They are known as rational Cherednik algebras (symplectic reflection algebras) and finite W-algebras. Losev showed that these two quantizations are isomor- phic by using realization of these algebras in terms of deformation-quantization. One can define a cohomology theory associated with Hamiltonian reduction, which is known as BRST cohomologies. In this talk, we see that higher BRST cohomologies corre- sponding to the rational Cherednik algebras do not vanish, while ones corresponding to the finite W-algebras vanish. Moreover, we see that the higher BRST cohomologies can be determined explicitly. To determine the higher cohomologies, we use the real- ization as deformation-quantization algebras and affinity properties of these sheaves of deformation-quantization algebras.