There is a couple of papers on this [Braverman, Maulik, Oblomkov, Okounkov, Pandharipande] where authors calculate quantum cohomology for various conical symplectic resolutions. The language in these paper is pretty algebro-geometric, whereas my view of quantum cohomology comes from symplectic geometry. Therefore, instead of ''counting rational curves'' I think of counting holomorphic=symplectic spheres. 


Though, in the exact setup as this is (holomorphic symplectic form in conical symplectic resolutions is exact) there are no nontrivial holomorphic curves. So the quantum cohomology = singular cohomology as a ring.


However, in these papers, authors deal with equivariant quantum cohomology rather than the usual one (these spaces always come with a group action). So, as a vector space it is just equivariant cohomology tensored with appropriate power series ring, but I remain puzzled with the **ring multiplication in it** - thinking from symplectic viewpoint, does it counts some sort of ''equivariant holomorphic = symplectic spheres''?



Equivariant quantum cohomology of conical symplectic resolutions
First of all, I don't understand why you say that there are no holomorphic curves - a typical example of such a space is $T^\*{\mathbb C}{\mathbb P}^1$ and it perfectly has non-trivial holomoprhic curves. What is true though, is that you can choose a different complex structure (still compatible with the same real symplectic form) so that all holomorphic curves go away. That's one of the possible explanations why on usual (non-equivariant) quantum cohomology the multiplication is equal to the ordinary (non-quantum) multiplication. Hoewever, the point is that there is no complex (or even almost complex) structure with this property in which the ${\mathbb C}^\*$-action remains holomoprhic. Basically, the correct way to think about equivariant cohomology (in any context) is to think about cohomology of the quotient. Of course, this is tricky when the action is not free, but philosophically that's what it is. So, you have to work with complex structures which are invariant under ${\mathbb C}^\*$, and then you do have curves. There is still an issue of defining the right virtual fundamental class on the corresponding moduli space. 


