  We investigate the application of matrix product state (MPS) representations
of the influence functionals (IF) for the calculation of real-time equilibrium
correlation functions in open quantum systems. Focusing specifically on the
unbiased spin-boson model, we explore the use of IF-MPSs for complex time
propagation, as well as IF-MPSs for constructing correlation functions in the
steady state. We examine three different IF approaches: one based on the
Kadanoff-Baym contour targeting correlation functions at all times, one based
on a complex contour targeting the correlation function at a single time, and a
steady state formulation which avoids imaginary or complex times, while
providing access to correlation functions at all times. We show that within the
IF language, the steady state formulation provides a powerful approach to
evaluate equilibrium correlation functions.
