  We theoretically investigate the propagation of heat currents in a
three-terminal quantum dot engine. Electron-electron interactions introduce
state-dependent processes which can be resolved by energy-dependent tunneling
rates. We identify the relevant transitions which define the operation of the
system as a thermal transistor or a thermal diode. In the former case,
thermal-induced charge fluctuations in the gate dot modify the thermal currents
in the conductor with suppressed heat injection, resulting in huge
amplification factors and the possible gating with arbitrarily low energy cost.
In the latter case, enhanced correlations of the state-selective tunneling
transitions redistribute heat flows giving high rectification coefficients and
the unexpected cooling of one conductor terminal by heating the other one. We
propose quantum dot arrays as a possible way to achieve the extreme tunneling
asymmetries required for the different operations.
