Correlations in photodetection signals from quantum light sources are conventionally calculated by application of the source master equation and the quantum regression theorem. We explore the temporal correlation functions in a single atom, two atoms with interacting Rydberg states, and a cavity quantum electrodynamics system. The transient correlation of either the field-amplitude or the intensity measurements with the later photon counting events can be well understood in terms of a recently developed theory of past quantum states of a monitored quantum system.