  The round complexity of interactive proof systems is a key question of
practical and theoretical relevance in complexity theory and cryptography.
Moreover, results such as QIP = QIP(3) (STOC'00) show that quantum resources
significantly help in such a task.
  In this work, we initiate the study of round compression of protocols in the
bounded quantum storage model (BQSM). In this model, the malicious parties have
a bounded quantum memory and they cannot store the all the qubits that are
transmitted in the protocol.
  Our main results in this setting are the following:
  1. There is a non-interactive (statistical) witness indistinguishable proof
for any language in NP (and even QMA) in BQSM in the plain model. We notice
that in this protocol, only the memory of the verifier is bounded.
  2. Any classical proof system can be compressed in a two-message quantum
proof system in BQSM. Moreover, if the original proof system is zero-knowledge,
the quantum protocol is zero-knowledge too. In this result, we assume that the
prover has bounded memory.
  Finally, we give evidence towards the "tightness" of our results. First, we
show that NIZK in the plain model against BQS adversaries is unlikely with
standard techniques. Second, we prove that without the BQS model there is no
2-message zero-knowledge quantum interactive proof, even under computational
assumptions.
