  We initiate the study of non-interactive zero-knowledge (NIZK) arguments for
languages in QMA. Our first main result is the following: if Learning With
Errors (LWE) is hard for quantum computers, then any language in QMA has an
NIZK argument with preprocessing. The preprocessing in our argument system
consists of (i) the generation of a CRS and (ii) a single
(instance-independent) quantum message from verifier to prover. The
instance-dependent phase of our argument system involves only a single
classical message from prover to verifier. Importantly, verification in our
protocol is entirely classical, and the verifier needs not have quantum memory;
its only quantum actions are in the preprocessing phase. Our second
contribution is to extend the notion of a classical proof of knowledge to the
quantum setting. We introduce the notions of arguments and proofs of quantum
knowledge (AoQK/PoQK), and we show that our non-interactive argument system
satisfies the definition of an AoQK. In particular, we explicitly construct an
extractor which can recover a quantum witness from any prover which is
successful in our protocol. Finally, we show that any language in QMA has an
(interactive) proof of quantum knowledge.
