  Many-body localized (MBL) phases of disordered quantum many-particle systems
have a number of unique properties, including failure to act as a thermal bath
and protection of quantum coherence. Studying MBL is complicated by the effects
of rare ergodic regions, necessitating large system sizes and averaging over
many disorder configurations. Here, building on the Feynman-Vernon theory of
quantum baths, we characterize the quantum noise that a disordered spin system
exerts on its parts via an influence matrix (IM). In this approach, disorder
averaging is implemented exactly, and the thermodynamic-limit IM obeys a
self-consistency equation. Viewed as a wavefunction in the space of
trajectories of an individual spin, the IM exhibits slow scaling of temporal
entanglement in the MBL phase. This enables efficient matrix product states
computations to obtain temporal correlations, providing a benchmark for quantum
simulations of non-equilibrium matter. The IM quantum noise formulation
provides an alternative starting point for novel rigorous studies of MBL.
