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 $Ω(\log^*λ)$ 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.