  We investigate the existence of constant-round post-quantum black-box
zero-knowledge protocols for $\mathbf{NP}$. As a main result, we show that
there is no constant-round post-quantum black-box zero-knowledge argument for
$\mathbf{NP}$ unless $\mathbf{NP}\subseteq \mathbf{BQP}$. As constant-round
black-box zero-knowledge arguments for $\mathbf{NP}$ exist in the classical
setting, our main result points out a fundamental difference between
post-quantum and classical zero-knowledge protocols. Combining previous
results, we conclude that unless $\mathbf{NP}\subseteq \mathbf{BQP}$,
constant-round post-quantum zero-knowledge protocols for $\mathbf{NP}$ exist if
and only if we use non-black-box techniques or relax certain security
requirements such as relaxing standard zero-knowledge to
$\epsilon$-zero-knowledge. Additionally, we also prove that three-round and
public-coin constant-round post-quantum black-box $\epsilon$-zero-knowledge
arguments for $\mathbf{NP}$ do not exist unless $\mathbf{NP}\subseteq
\mathbf{BQP}$.
