  Describing non-equilibrium properties of quantum many-body systems is
challenging due to high entanglement in the wavefunction. We describe evolution
of local observables via the influence matrix (IM), which encodes the effects
of a many-body system as an environment for local subsystems. Recent works
found that in many dynamical regimes the IM of an infinite system has low
temporal entanglement and can be efficiently represented as a matrix-product
state (MPS). Yet, direct iterative constructions of the IM encounter highly
entangled intermediate states - a temporal entanglement barrier (TEB). We argue
that TEB is ubiquitous, and elucidate its physical origin via a semiclassical
quasiparticle picture that exactly captures the behavior of integrable spin
chains. Further, we show that a TEB also arises in chaotic spin chains, which
lack well-defined quasiparticles. Based on these insights, we formulate an
alternative light-cone growth algorithm, which provably avoids TEB, thus
providing an efficient construction of the thermodynamic-limit IM as a MPS.
This work uncovers the origin of the efficiency of the IM approach for
thermalization and transport.
