  The concept of space-evolution (or space-time duality) has emerged as a
promising approach for studying quantum dynamics. The basic idea involves
exchanging the roles of space and time, evolving the system using a space
transfer matrix rather than the time evolution operator. The infinite-volume
limit is then described by the fixed points of the latter transfer matrix, also
known as influence matrices. To establish the potential of this method as a
bona fide computational scheme, it is important to understand whether the
influence matrices can be efficiently encoded in a classical computer. Here we
begin this quest by presenting a systematic characterisation of their
entanglement -- dubbed temporal entanglement -- in chaotic quantum systems. We
consider the most general form of space-evolution, i.e., evolution in a generic
space-like direction, and present two fundamental results. First, we show that
temporal entanglement always follows a volume law in time. Second, we identify
two marginal cases -- (i) pure space evolution in generic chaotic systems (ii)
any space-like evolution in dual-unitary circuits -- where R\'enyi entropies
with index larger than one are sub-linear in time while the von Neumann
entanglement entropy grows linearly. We attribute this behaviour to the
existence of a product state with large overlap with the influence matrices.
This unexpected structure in the temporal entanglement spectrum might be the
key to an efficient computational implementation of the space evolution.
