  A semiconductor quantum dot (QD) embedded within an optical microcavity is a
system of fundamental importance within quantum information processing. The
optimization of quantum coherence is crucial in such applications, requiring an
in-depth understanding of the relevant decoherence mechanisms. We provide
herein a critical review of prevalent theoretical treatments of the QD-cavity
system coupled to longitudinal acoustic phonons, comparing predictions against
a recently obtained exact solution. Within this review we consider a range of
temperatures and exciton-cavity coupling strengths. Predictions of the polaron
Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equations, as well
as a variation of the former adapted for adiabatic continuous wave excitation
(CWE), are compared against an asymptotically exact solution based upon
Trotter's decomposition (TD) theorem. The NZ and TCL implementations, which
apply a polaron transformation to the Hamiltonian and subsequently treat the
exciton-cavity coupling to second order, do not offer a significant improvement
accuracy relative to the polaron transformation alone. The CWE adaptation
provides a marked improvement, capturing the broadband features of the
absorption spectrum (not present in NZ and TCL implementations). We attribute
this difference to the effect of the Markov approximation, and particularly its
unsuitability in pulsed excitation regime. Even the CWE adaptation, however,
breaks down in the regime of high temperature ($50K$) and strong exciton-cavity
coupling ($g \gtrsim 0.2$ meV). The TD solution is of comparable computational
complexity to the above-mentioned master equation approaches, yet remains
accurate at higher temperatures and across a broad range of exciton-cavity
coupling strengths (at least up to $g=1.5$ meV).
