  We present two different methods for modelling non-Markovian quantum
light-matter interactions in waveguide QED systems, using matrix product states
(MPSs) and a space-discretized waveguide (SDW) model. After describing the
general theory and implementation of both approaches, we compare and contrast
these methods directly on three topical problems of interest in waveguide-QED,
including (i) a two-level system (TLS) coupled to an infinite (one-dimensional)
waveguide, (ii) a TLS coupled to a terminated waveguide with a time-delayed
coherent feedback, and (iii) two spatially separated TLSs coupled within an
infinite waveguide. Both approaches are shown to efficiently model multi-photon
nonlinear dynamics in highly non-Markovian regimes, and we highlight the
advantages and disadvantages of these methods for modelling waveguide QED
interactions, including their implementation in Python, computational run
times, and ease of conceptual understanding. We explore both vacuum dynamics as
well as regimes of strong optical pumping, where a weak excitation
approximation cannot be applied. The MPS approach scales better when modelling
multi-photon dynamics and long delay times, and explicitly includes
non-Markovian memory effects. In contrast, the SDW model accounts for
non-Markovian effects through space discretization, and solves Markovian
equations of motion, yet rigorously includes the effects of retardation. The
SDW model, based on an extension of recent collisional pictures in quantum
optics, is solved through quantum trajectory techniques, and can more easily
add in additional dissipation processes, including off-chip decay and TLS pure
dephasing. The impact of these processes is shown directly on feedback-induced
population trapping and TLS entanglement between spatially separated TLSs.
