Quantum-cascade lasers are provided with an active section in which relaxation of carriers from a lower laser level is provided by three or more phonon-assisted transitions to levels within the active section whose energies are below the energy of the lower laser level. The gain region of the laser consists of alternating active and injector sections, with an injection barrier inserted between each injector section and the adjacent active section, and an exit barrier inserted between each active section and the adjacent injector section. The active section comprises a sufficient number of quantum wells separated by quantum barriers to produce the desired energy-level structure consisting of an upper laser level, a lower laser level, and at least three levels that have lower energies than the lower laser level, with the separation of adjacent energy levels below and including the lower laser level that are at least equal to the energy of the quantum well material's longitudinal optical phonon.