  We provide the first $\mathit{constant}$-$\mathit{round}$ construction of
post-quantum non-malleable commitments under the minimal assumption that
$\mathit{post}$-$\mathit{quantum}$ $\mathit{one}$-$\mathit{way}$
$\mathit{functions}$ exist. We achieve the standard notion of non-malleability
with respect to commitments. Prior constructions required
$\Omega(\log^*\lambda)$ rounds under the same assumption.
  We achieve our results through a new technique for constant-round
non-malleable commitments which is easier to use in the post-quantum setting.
The technique also yields an almost elementary proof of security for
constant-round non-malleable commitments in the classical setting, which may be
of independent interest.
  When combined with existing work, our results yield the first constant-round
quantum-secure multiparty computation for both classical and quantum
functionalities $\mathit{in}$ $\mathit{the}$ $\mathit{plain}$ $\mathit{model}$,
under the $\mathit{polynomial}$ hardness of quantum fully-homomorphic
encryption and quantum learning with errors.
