Neuromorphic in-memory computing systems, comprising artificial synapses and neurons, can overcome the energy inefficiency and throughput limitation of today's von Neumann computing architecture. Recently, powered by the unique properties of quantum materials, for example, high mobility, outstanding sensitivity, and strong quantum effect, researchers have built quantum artificial synapses to mimic the biological ones. These quantum electronic/photonic synapses can precisely define their conductance state (or synaptic weight) for emulating synaptic behaviors, which shows bionic performance unreachable by other conventional materials. In this review, the significant achievements in quantum artificial synapses are summarized. First, potential quantum materials used in artificial synapses are discussed with particular attention to quantum dots, nanowires, layered materials, and quasi-2DEG interfaces. Then, the major quantum effects that are utilized in quantum artificial synapses, for example, Josephson effect, quantum tunneling, and spin memory, are reviewed. In addition to the discussion on a single synaptic device, the macroscale integration into artificial visual systems and artificial nerve networks are also highlighted. Finally, the associated future research trends and target applications are also discussed.