Recently, solutions of the Ishibashi, Kawai, Kitazawa and Tsuchiya matrix theory have been found, which can be interpreted as 3+1-dimensional quantum geometries describing an effective Friedmann-Lema\^{i}tre-Robertson-Walker cosmology with a big bounce. In this paper, we examine the propagation of a scalar field in an open Friedmann-Lema\^{i}tre-Robertson-Walker spacetime arising within this framework. The paper is divided into two parts. In the first one, we perform a classical investigation by resorting to general-relativity tools where we show that both massless and massive non-interacting particles can travel across the big bounce. In the second part, we evaluate the scalar field propagator by means of quantum-field-theory techniques. This analysis reveals that in the late-time regime the scalar propagator resembles the standard Feynman propagator of flat Minkowski space, whereas for early times it gives rise to a well-defined correlation between two points on opposite sheets of the spacetime. The paper is based on Ref. \cite{Battista:2022hqn}.