Let $X$ be the total space of $\mathscr{O}(-1)^{\oplus{n}}\rightarrow\mathbb{P}^m$. I think there should be some general way to compute its quantum cohomology $QH^\ast(X)$.

However, since I'm not familiar with these things, I found it hard to carry out the computations directly by hand without assuming that $m\gg n$, or both of $m$ and $n$ are very small, and of course it's trivial when $n=1$. When $m\gg n$, by checking dimensions one sees easily that the degree of the holomorphic curves must be 0 or 1 in order to obtain non-trivial Gromov-Witten invariants, but this simplification does not seem to happen in general.

Is there any coherent way to compute $QH^\ast(X)$ for all the possible $m,n\geq1$? I think the problem should be trivial to experts and it's quite possible that it can be found in the literature.