  We propose three constructions of classically verifiable non-interactive
zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing
models.
  - We construct a CV-NIZK for QMA in the quantum secret parameter model where
a trusted setup sends a quantum proving key to the prover and a classical
verification key to the verifier. It is information theoretically sound and
zero-knowledge.
  - Assuming the quantum hardness of the learning with errors problem, we
construct a CV-NIZK for QMA in a model where a trusted party generates a CRS
and the verifier sends an instance-independent quantum message to the prover as
preprocessing. This model is the same as one considered in the recent work by
Coladangelo, Vidick, and Zhang (CRYPTO '20).
  Our construction has the so-called dual-mode property, which means that there
are two computationally indistinguishable modes of generating CRS, and we have
information theoretical soundness in one mode and information theoretical
zero-knowledge property in the other. This answers an open problem left by
Coladangelo et al, which is to achieve either of soundness or zero-knowledge
information theoretically. To the best of our knowledge, ours is the first
dual-mode NIZK for QMA in any kind of model.
  - We construct a CV-NIZK for QMA with quantum preprocessing in the quantum
random oracle model. This quantum preprocessing is the one where the verifier
sends a random Pauli-basis states to the prover. Our construction uses the
Fiat-Shamir transformation. The quantum preprocessing can be replaced with the
setup that distributes Bell pairs among the prover and the verifier, and
therefore we solve the open problem by Broadbent and Grilo (FOCS '20) about the
possibility of NIZK for QMA in the shared Bell pair model via the Fiat-Shamir
transformation.
