AbstractMemristors are resistive elements retaining information of their past dynamics. They have garnered substantial interest due to their potential for representing a paradigm change in electronics, information processing and unconventional computing. Given the advent of quantum technologies, a design for a quantum memristor with superconducting circuits may be envisaged. Along these lines, we introduce such a quantum device whose memristive behavior arises from quasiparticle-induced tunneling when supercurrents are cancelled. For realistic parameters, we find that the relevant hysteretic behavior may be observed using current state-of-the-art measurements of the phase-driven tunneling current. Finally, we develop suitable methods to quantify memory retention in the system.