Hi, I know very little about the quantum cohomology (QC for short). I only got interested in the subject as the genus zero part may be relevant to a problem I'm working on. So I hope my question makes sense.

I understand QC defines a structure of an algebra over the operad $H^*(\bar{M}_{g,n})$ on the cohomology $H^*(V)$ of any smooth projective complex variety $V$.

In Hodge theory, the yoga of weight filtration extends enriched structure on the cohomology from smooth projective varieties to smooth varieties. This is done by using resolution of singularities to represent a quasi-projective smooth variety $X$ as the complement $\bar{X} \setminus D$ of a normal crossing divisor in a projective variety. Then one can compute the cohomology $H^* (X)$ in terms of the cohomology $H^* (D_q)$ of the closed stata using a spectral sequence.

Question: Is there such a thing for quantum cohomology?

I'm not sure functoriality of QC is established except for automorphism. Is that the only obstacle?