In mathematical challenges from theoretical / computational chemistry by Committee on Mathematical Challenges from Computational Chemistry, "The N-and V-Representability Problems" is listed.
> Unfortunately, only an incomplete set of necessary conditions are known, but these are already so complex that further work in this area has been abandoned by chemists.
I think this is solved with negative result: computational complexity is QMA-complete.
If I remembered correctly Ayers et al. have already published proof.
Nakata Maho is a scientist and interested in reduced density matrix related theories, optimization and multiple precision arithmetics.