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We investigate the existence of constant-round post-quantum black-boxzero-knowledge protocols for $\mathbf{NP}$. As a main result, we show thatthere is no constant-round post-quantum black-box zero-knowledge argument for$\mathbf{NP}$ unless $\mathbf{NP}\subseteq \mathbf{BQP}$. As constant-roundblack-box zero-knowledge arguments for $\mathbf{NP}$ exist in the classicalsetting, our main result points out a fundamental difference betweenpost-quantum and classical zero-knowledge protocols. Combining previousresults, we conclude that unless $\mathbf{NP}\subseteq \mathbf{BQP}$,constant-round post-quantum zero-knowledge protocols for $\mathbf{NP}$ exist ifand only if we use non-black-box techniques or relax certain securityrequirements such as relaxing standard zero-knowledge to$\epsilon$-zero-knowledge. Additionally, we also prove that three-round andpublic-coin constant-round post-quantum black-box $\epsilon$-zero-knowledgearguments for $\mathbf{NP}$ do not exist unless $\mathbf{NP}\subseteq\mathbf{BQP}$.

On the Impossibility of Post-Quantum Black-Box Zero-Knowledge in Constant Rounds