  We study the notion of zero-knowledge secure against quantum polynomial-time
verifiers (referred to as quantum zero-knowledge) in the concurrent composition
setting. Despite being extensively studied in the classical setting, concurrent
composition in the quantum setting has hardly been studied. We initiate a
formal study of concurrent quantum zero-knowledge. Our results are as follows:
  -Bounded Concurrent QZK for NP and QMA: Assuming post-quantum one-way
functions, there exists a quantum zero-knowledge proof system for NP in the
bounded concurrent setting. In this setting, we fix a priori the number of
verifiers that can simultaneously interact with the prover. Under the same
assumption, we also show that there exists a quantum zero-knowledge proof
system for QMA in the bounded concurrency setting.
  -Quantum Proofs of Knowledge: Assuming quantum hardness of learning with
errors (QLWE), there exists a bounded concurrent zero-knowledge proof system
for NP satisfying quantum proof of knowledge property. Our extraction mechanism
simultaneously allows for extraction probability to be negligibly close to
acceptance probability (extractability) and also ensures that the prover's
state after extraction is statistically close to the prover's state after
interacting with the verifier (simulatability). The seminal work of [Unruh
EUROCRYPT'12], and all its followups, satisfied a weaker version of
extractability property and moreover, did not achieve simulatability. Our
result yields a proof of quantum knowledge system for QMA with better
parameters than prior works.
