  Semiconductor quantum dots have recently emerged as a leading platform to
efficiently generate highly indistinguishable photons, and this work addresses
the timely question of how good these solid-state sources can ultimately be. We
establish the crucial role of lattice relaxation in these systems in giving
rise to trade-offs between indistinguishability and efficiency. We analyse the
two source architectures most commonly employed: a quantum dot embedded in a
waveguide and a quantum dot coupled to an optical cavity. For waveguides, we
demonstrate that the broadband Purcell effect results in a simple inverse
relationship, where indistinguishability and efficiency cannot be
simultaneously increased. For cavities, the frequency selectivity of the
Purcell enhancement results in a more subtle trade-off, where
indistinguishability and efficiency can be simultaneously increased, though by
the same mechanism not arbitrarily, limiting a source with near-unity
indistinguishability ($>99$\%) to an efficiency of approximately 96\% for
realistic parameters.
