I think that if you take an affine variety all of its Gromov-Witten invariants of degree $\neq 0$ are $0$ in any sense. So QC for affine varieties should coincide with ordinary cohomology.
The following point of view might be useful: you might (and in fact, should) think about small quantum cohomology as some kind of Floer cohomology of the loop space of $X$. If you cover $X$ by subsets, then the loop space is NOT covered by the corresponding loop spaces, so your idea doesn't seem right to me...