  In this work we provide a complete description of the lifecycle of
entanglement during the real-time motion of open quantum systems. The quantum
environment can have arbitrary (e.g. structured) spectral density. The
entanglement can be seen constructively as a Lego: its bricks are the modes of
the environment. These bricks are connected to each other via operator
transforms. The central result is that each infinitesimal time interval one new
(incoming) mode of the environment gets coupled (entangled) to the open system,
and one new (outgoing) mode gets irreversibly decoupled (disentangled from
future). Moreover, each moment of time, only a few relevant modes (3 - 4 in the
considered cases) are non-negligibly coupled to the future quantum motion.
These relevant mode change (flow, or renormalize) with time. As a result, the
temporal entanglement has the structure of a matrix-product operator. This
allows us to pose a number of questions and to answer them in this work: what
is the intrinsic quantum complexity of a real time motion; does this complexity
saturate with time, or grows without bounds; how to do the real-time
renormalization group in a justified way; how the classical Brownian stochastic
trajectories emerge from the quantum evolution; how to construct the few-mode
representations of non-Markovian environments. We provide illustrative
simulations of the spin-boson model for various spectral densities of the
environment: semicircle, subohmic, Ohmic, and superohmic.
