On temporal entropy and the complexity of computing the expectation value of local operators after a quench

We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states. We argue that such cost is intimately related to that of encoding temporal transition matrices and their partial traces. In particular, we show that we can upper-bound the rank of these reduced transition matrices by the one of the Heisenberg evolution of local operators, thus making connection between two apparently different quantities, the temporal entanglement and the local operator entanglement. As a result, whenever the local operator entanglement grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal matrix product states is likely advantageous with respect to computing the same quantities using standard matrix product states techniques.