Superconductors are known to be excellent thermal insulators at low temperature owing to the presence of the energy gap in their density of states (DOS). In this context, the superconducting textit{proximity effect} allows to tune the local DOS of a metallic wire by controlling the phase bias ($varphi$) imposed across it. As a result, the wire thermal conductance can be tuned over several orders of magnitude by phase manipulation. Despite strong implications in nanoscale heat management, experimental proofs of phase-driven control of thermal transport in superconducting proximitized nanostructures are still very limited. Here, we report the experimental demonstration of efficient heat current control by phase tuning the superconducting proximity effect. This is achieved by exploiting the magnetic flux-driven manipulation of the DOS of a quasi one-dimensional aluminum nanowire forming a weal-link embedded in a superconducting ring. Our thermal superconducting quantum interference transistor (T-SQUIPT) shows temperature modulations up to $sim 16$ mK yielding a temperature-to-flux transfer function as large as $sim 60$ mK/$Phi_0$. Yet, phase-slip transitions occurring in the nanowire Josephson junction induce a hysteretic dependence of its local DOS on the direction of the applied magnetic field. Thus, we also prove the operation of the T-SQUIPT as a phase-tunable textit{thermal memory}, where the information is encoded in the temperature of the metallic mesoscopic island. Besides their relevance in quantum physics, our results are pivotal for the design of innovative coherent caloritronics devices such as heat valves and temperature amplifiers suitable for thermal logic architectures.
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