  We propose three constructions of classically verifiable non-interactive
proofs (CV-NIP) and non-interactive zero-knowledge proofs and arguments
(CV-NIZK) for QMA in various preprocessing models.
  - We construct an information theoretically sound CV-NIP for QMA in the
secret parameter model where a trusted party generates a quantum proving key
and classical verification key and gives them to the corresponding parties
while keeping it secret from the other party. Alternatively, we can think of
the protocol as one in a model where the verifier sends an instance-independent
quantum message to the prover as preprocessing.
  - We construct a CV-NIZK for QMA in the secret parameter model. It is
information theoretically sound and zero-knowledge.
  - Assuming the quantum hardness of the leaning 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.
