In this work, we derive the first lifting theorems for establishing security in the quantum random permutation and ideal cipher models. These theorems relate the success probability of an arbitrary quantum adversary to that of a classical algorithm making only a small number of classical queries. By applying these lifting theorems, we improve previous results and obtain new quantum query complexity bounds and post-quantum security results. Notably, we derive tight bounds for the quantum hardness of the double-sided zero search game and establish the post-quantum security for the preimage resistance, one-wayness, and multi-collision resistance of constant-round sponge, as well as the collision resistance of the Davies-Meyer construction.