  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.
