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：我们研究了一个多项式时间判定问题，其中每个输入编码了一次深度为N的因果执行过程，在该过程中单个不可复制的令牌必须遍历一个有序步骤序列，且每一步最多暴露O(1)位的路由信息。该问题的不确定性在于通过中继网络识别传递路径，而非最终的接受/拒绝结果——后者仅由预设执行的完成来定义。确定性图灵机以Θ(N)时间执行该过程。利用信息论工具——具体而言是中继信道的割集界和Fano不等式——我们证明了任何遵循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，不存在经典的NC电路族能够实现该过程。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。