Memristive systems are generalizations of memristors, which are resistors with memory. In this paper, we present a quantum description of quantum dot memristive systems. Using this model we propose and experimentally demonstrate a simple and practical scheme for realizing memristive systems with quantum dots. The approach harnesses a phenomenon that is commonly seen as a bane of nanoelectronics, i.e., switching of a trapped charge in the vicinity of the device. We show that quantum dot memristive systems have hysteresis current-voltage characteristics and quantum jump-induced stochastic behavior. While our experiment requires low temperatures, the same setup could, in principle, be realized with a suitable single-molecule transistor and operated at or near room temperature.