We investigate the evolution of a target qubit caused by its multiple random collisions with N-qubit clusters. Depending on the cluster state, the evolution of the target qubit may correspond to its effective interaction with a thermal bath, a coherent (laser) drive, or a squeezed bath. In cases where the target qubit relaxes to a thermal state, its dynamics can exhibit a quantum advantage, whereby the target-qubit temperature can be scaled up proportionally to N^{2} and the thermalization time can be shortened by a similar factor, provided the appropriate coherence in the cluster is prepared by nonthermal means. We dub these effects quantum superthermalization because of the analogies to superradiance. Experimental realizations of these effects are suggested.