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 $mathbf{NP}subseteq mathbf{BQP}$.
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