We give the first proof of security for an identity-based encryption (IBE) scheme in the quantum random oracle model. This is the first proof of security for any scheme in this model that does not rely on the assumed existence of so-called quantum-secure pseudorandom functions (PRFs). Our techniques are quite general and we use them to obtain security proofs for two random oracle hierarchical IBE schemes and a random oracle signature scheme, all of which have previously resisted quantum security proofs, even assuming quantum-secure PRFs. We also explain how to remove quantum-secure PRFs from prior quantum random oracle model proofs. We accomplish these results by developing new tools for arguing that quantum algorithms cannot distinguish between two oracle distributions. Using a particular class of oracle distributions that we call semi-constant distributions, we argue that the aforementioned cryptosystems are secure against quantum adversaries.