Is it possible to convert classical cryptographic reductions into post-quantum ones? It is customary to argue that while this is problematic in the interactive setting, non-interactive reductions do carry over. However, when considering quantum auxiliary input, this conversion results in a non-constructive post-quantum reduction that requires duplicating the quantum auxiliary input, which is in general inefficient or even impossible. This violates the win-win premise of provable cryptography: an attack against a cryptographic primitive should lead to an algorithmic advantage. We initiate the study of constructive quantum reductions and present positive and negative results for converting large classes of classical reductions to the post-quantum setting in a constructive manner. We show that any non-interactive non-adaptive reduction from assumptions with a polynomial solution space (such as decision assumptions) can be made post-quantum constructive. In contrast, assumptions with super-polynomial solution space (such as general search assumptions) cannot be generally converted. Along the way, we make several additional contributions: 1. We put forth a framework for reductions (or general interaction) with stateful solvers for a computational problem, that may change their internal state between consecutive calls. We show that such solvers can still be utilized. This framework and our results are meaningful even in the classical setting. 2. A consequence of our negative result is that quantum auxiliary input that is useful against a problem with a super-polynomial solution space cannot be generically ``restored'' post-measurement. This shows that the novel rewinding technique of Chiesa et al. (FOCS 2021) is tight in the sense that it cannot be extended beyond a polynomial measurement space.