  We construct the first constant-round protocols for secure quantum
computation in the two-party (2PQC) and multi-party (MPQC) settings with
security against malicious adversaries. Our protocols are in the common random
string (CRS) model.
  - Assuming two-message oblivious transfer (OT), we obtain (i) three-message
2PQC, and (ii) five-round MPQC with only three rounds of online
(input-dependent) communication; such OT is known from quantum-hard Learning
with Errors (QLWE).
  - Assuming sub-exponential hardness of QLWE, we obtain (i) three-round 2PQC
with two online rounds and (ii) four-round MPQC with two online rounds.
  - When only one (out of two) parties receives output, we achieve minimal
interaction (two messages) from two-message OT; classically, such protocols are
known as non-interactive secure computation (NISC), and our result constitutes
the first maliciously-secure quantum NISC. Additionally assuming reusable
malicious designated-verifier NIZK arguments for NP (MDV-NIZKs), we give the
first MDV-NIZK for QMA that only requires one copy of the quantum witness.
  Finally, we perform a preliminary investigation into two-round secure quantum
computation where each party must obtain output. On the negative side, we
identify a broad class of simulation strategies that suffice for classical
two-round secure computation that are unlikely to work in the quantum setting.
Next, as a proof-of-concept, we show that two-round secure quantum computation
exists with respect to a quantum oracle.
