  From the minimal assumption of post-quantum semi-honest oblivious transfers,
we build the first $\epsilon$-simulatable two-party computation (2PC) against
quantum polynomial-time (QPT) adversaries that is both constant-round and
black-box (for both the construction and security reduction). A recent work by
Chia, Chung, Liu, and Yamakawa (FOCS'21) shows that post-quantum 2PC with
standard simulation-based security is impossible in constant rounds, unless
either $\mathbf{NP} \subseteq \mathbf{BQP}$ or relying on non-black-box
simulation. The $\epsilon$-simulatability we target is a relaxation of the
standard simulation-based security that allows for an arbitrarily small
noticeable simulation error $\epsilon$. Moreover, when quantum communication is
allowed, we can further weaken the assumption to post-quantum secure one-way
functions (PQ-OWFs), while maintaining the constant-round and black-box
property.
  Our techniques also yield the following set of constant-round and black-box
two-party protocols secure against QPT adversaries, only assuming black-box
access to PQ-OWFs:
  - extractable commitments for which the extractor is also an
$\epsilon$-simulator;
  - $\epsilon$-zero-knowledge commit-and-prove whose commit stage is
extractable with $\epsilon$-simulation;
  - $\epsilon$-simulatable coin-flipping;
  - $\epsilon$-zero-knowledge arguments of knowledge for $\mathbf{NP}$ for
which the knowledge extractor is also an $\epsilon$-simulator;
  - $\epsilon$-zero-knowledge arguments for $\mathbf{QMA}$.
  At the heart of the above results is a black-box extraction lemma showing how
to efficiently extract secrets from QPT adversaries while disturbing their
quantum state in a controllable manner, i.e., achieving
$\epsilon$-simulatability of the post-extraction state of the adversary.
