  Optical nonlinearities are known to provide a coherent coupling between the
amplitude and phase of the light, which can result in the formation of periodic
waveforms. Lasers that emit such waveforms are referred to as optical frequency
combs. Here we show that Bloch gain - a nonclassical phenomenon that was first
predicted in the 1930s - plays an essential role in comb formation in quantum
cascade lasers (QCLs). We develop a self-consistent theoretical model that
considers all aspects of comb formation: bandstructure, electron transport, and
cavity dynamics. It reveals that Bloch gain gives rise to a giant Kerr
nonlinearity and serves as the physical origin of the linewidth enhancement
factor in QCLs. Using a master equation approach, we explain how frequency
modulated combs can be produced in Fabry-P\'{e}rot QCLs over the entire bias
range. In ring resonators, Bloch gain triggers phase turbulence and the
formation of soliton-like patterns.
