We study a polynomial-time decision problem in which each input encodes a depth-$N$ causal execution in which a single non-duplicable token must traverse an ordered sequence of steps, revealing at most $O(1)$ bits of routing information at each step. The uncertainty in the problem lies in identifying the delivery path through the relay network rather than in the final accept/reject outcome, which is defined solely by completion of the prescribed execution. A deterministic Turing machine executes the process in $Θ(N)$ time. Using information-theoretic tools - specifically cut-set bounds for relay channels and Fano's inequality - we prove that any execution respecting the causal constraints requires $Ω(N)$ units of causal time, thereby ruling out asymptotic parallel speedup. We further show that no classical $\mathbf{NC}$ circuit family can implement the process when circuit depth is interpreted as realizable parallel time. This identifies a class of polynomial-time problems with intrinsic causal structure and highlights a gap between logical parallelism and causal executability.

翻译：我们研究一个多项式时间判定问题，其中每个输入编码一个深度为 $N$ 的因果执行过程：一个不可复制的令牌必须遍历一个有序的步骤序列，且每一步最多只能揭示 $O(1)$ 比特的路由信息。该问题的不确定性在于识别通过中继网络的传递路径，而非最终的接受/拒绝结果（该结果仅由预定执行的完成情况定义）。一台确定性图灵机以 $Θ(N)$ 的时间执行该过程。利用信息论工具——特别是中继信道的割集界和法诺不等式——我们证明了任何遵守因果约束的执行都需要 $Ω(N)$ 单位的因果时间，从而排除了渐近并行加速的可能性。我们进一步证明，当电路深度被解释为可实现的并行时间时，没有经典的 $\mathbf{NC}$ 电路族可以实现该过程。这识别了一类具有内在因果结构的多项式时间问题，并凸显了逻辑并行性与因果可执行性之间的差距。