Let $X$ be a $2n$-dimensional symplectic manifold or a smooth $n$-dimensional algebraic variety over $\mathbb{C}$, which is not necessarily compact, but as part of the requirement the small quantum cohomology $\mathit{QH}^\ast(X)$ should be well-defined.
Now consider the $k$-th quantum power of the first Chern class $c_1(X)^{\star k}$, which necessarily lies in $\mathit{QH}^{2k}(X)$, where the grading on $\mathit{QH}^\ast(X)$ is understood in a reduced way, by moding $2N$, with $N$ being the minimal Chern number.
My question is when will there be a $k$ with $\lceil n/2\rceil\leq k\leq n-1$ such that $c_1(X)^{\star k}$ lies in the summand $H^{2k}(X;\mathbb{C})\subset\mathit{QH}^{2k}(X)$? Is there a convenient criterion which ensures that this holds?
As a concrete example, take $X$ to be the total space of the tautological line bundle $\mathcal{O}(-1)$ over $\mathbb{CP}^{n-1}$, then for any $k$ with $1\leq k\leq n-1$, one can check that the quantum power $c_1(X)^{\star k}$ equals the exterior power $c_1(M)^k$.