  In the path integral formulation of the evolution of an open quantum system
coupled to a Gaussian, non-interacting environment, the dynamical contribution
of the latter is encoded in an object called the influence functional. Here, we
relate the influence functional to the process tensor -- a more general
representation of a quantum stochastic process -- describing the evolution. We
then use this connection to motivate a tensor network algorithm for the
simulation of multi-time correlations in open systems, building on recent work
where the influence functional is represented in terms of time evolving matrix
product operators. By exploiting the symmetries of the influence functional, we
are able to use our algorithm to achieve orders-of-magnitude improvement in the
efficiency of the resulting numerical simulation. Our improved algorithm is
then applied to compute exact phonon emission spectra for the spin-boson model
with strong coupling, demonstrating a significant divergence from spectra
derived under commonly used assumptions of memorylessness.
