Lasers are recognized for coherent light emission, the onset of which is reflected in a change in the photon statistics. For many years, attempts have been made to directly measure correlations in the individual photon emission events of semiconductor lasers. Previously, the temporal decay of these correlations below or at the lasing threshold was considerably faster than could be measured with the time resolution provided by the Hanbury Brown/Twiss measurement set-up used. Here we demonstrate a measurement technique using a streak camera that overcomes this limitation and provides a record of the arrival times of individual photons. This allows us to investigate the dynamical evolution of correlations between the individual photon emission events. We apply our studies to micropillar lasers with semiconductor quantum dots as the active material, operating in the regime of cavity quantum electrodynamics. For laser resonators with a low cavity quality factor, Q, a smooth transition from photon bunching to uncorrelated emission with increasing pumping is observed; for high-Q resonators, we see a non-monotonic dependence around the threshold where quantum light emission can occur. We identify regimes of dynamical anti-bunching of photons in agreement with the predictions of a microscopic theory that includes semiconductor-specific effects.