  We propose a heat valve based on the interplay between thermal transport and
proximity-induced exchange splitting in Josephson tunnel junctions. We
demonstrate that the junction heat conductance strongly depends on the relative
alignment of the exchange fields induced in the superconductors. Colossal
magneto-heat resistance ratios as large as 10^7% are predicted to occur under
proper temperature and phase conditions, as well as suitable
ferromagnet-superconductor combinations. Moreover, the quantum phase tailoring,
intrinsic to the Josephson coupling, offers an additional degree of freedom for
the control of the heat conductance. Our predictions for the phase-coherent and
spin-dependent tuning of the thermal flux can provide a useful tool for heat
management at the nanoscale.
