Let $X$ and $Y$ manifolds. What kind of relations between them (like homeomorphism, diffeomorphism, homotopy equivalence) gives an isomorphic quantum cohomology rings?

Quantum cohomology, and more generally Gromov Witten invariants, are invariants under deformation. Given a family $X$ of compact symplectic manifolds over a base $B$, to a path in $B$ one can associate an isomorphism between the quantum cohomology of the fiber $X_0$ over the starting pooint of the path and the qc of the fiber $X_1$ over the end point of the path. Homotopic paths (homotopic wrt end points) induce the same isomorphism. In particular $\pi_1(B,0)$ acts on the quantum cohomology of $X_0$. 


If $X$ and $Y$ are CalabiYau threefolds which are obtained from each other from a simple flop, then their quantum cohomologies are isomorphic (Li and Ruan proved this). LeeLinWang prove the analog of this in higher dimensions. One of the things that makes this interesting is that the classical cohomology rings are not (in general) isomorphic, but the quantum cohomology rings are. More generally, if $X$ and $Y$ are birational and $K$equivalent (i.e. there is a resolution of the birational map $X\leftarrow W \to Y$ such that $K_X$ and $K_Y$ are isomorphic when pulled back to $W$), one expects their GromovWitten theories to be "equivalent". In some situations, "equivalent" will imply isomorphic quantum cohomologies, but not in general. To formulate the general equivalence properly requires Coates and Givental's Lagrangian cone formalism. Crepant resolutions are a special case of this and so the "crepant resolution conjecture" is part of this set of equivalences. Here are some papers on the subject (please forgive my laziness at only putting arXiv numbers and not journal references): LiRuan arXiv:math/9803036 BryanGraber arXiv:math/0610129 Iritani arXiv:0809.2749 LeeLinWang math.AG/0608370 CoatesRuan arXiv:0710.5901 


I am not an expert in the field, but I guess that there are examples of diffeomorphic symplectic manifolds with nonisomorphic quantum cohomologies (hence homotopy equivalence, homeomorphism, or even diffeomorphism alone is not enough to give an isomorphism on quantum cohomology). Two simply connected smooth 4manifolds having the same intersection form are homotopy equivalent by a theorem of Milnor. More than that, they are homeomorphic by a famous theorem of Freedman. However, very often these manifolds are not diffeomorphic. An example of homeomorphic but nondiffeomorphic pair is the Barlow surface and the blowup of the complex plane in 8 points. Note that both manifolds admit a Kahler structure. In the book of McDuff  Salamon Jholomorphic curves and quantum cohomology, they describe in Example 7.3.6 a construction due to Ruan of two diffeomorphic nondeformation equivalent 6manifolds. The manifolds are the two 4manifolds just described, both multiplied by $\mathbb{CP}^1$ (by a result of Wall, these two homeomorphic 4manifolds become diffeomorphic after such a stabilization). Ruan uses quantum cohmology to prove that these two diffeomorphic symplectic 6manifolds are not deformationequivalent, so my guess is that he uses that their quantum cohomologies are not isomorphic as a tool. 

