  The quantum regression theorem (QRT) is the most-widely used tool for
calculating multitime correlation functions for the assessment of quantum
emitters. It is an approximate method based on a Markov assumption for the
environmental coupling. In this work we quantify properties of photons emitted
from a single quantum dot coupled to phonons. For the single-photon purity and
the indistinguishability, we compare numerically exact path-integral results
with those obtained from the QRT. It is demonstrated that the QRT
systematically overestimates the influence of the environment for typical
quantum dots used in quantum information technology.
