Quantum Coherence Reshapes Thermodynamic Bounds for Thermal Machines

Thermodynamic Uncertainty Relations (TURs) set universal bounds linking current fluctuations to entropy production in nonequilibrium steady states. Their multidimensional generalization (MTUR) introduces matrix inequalities connecting current covariances and mean values. We analyze these bounds in a paradigmatic quantum thermal device, a two-terminal conductor, operating as a heat engine, refrigerator, or heat pump. We show that classical performance limits on efficiency and coefficient of performance remain constrained by the TUR when finite power or heat flow from cold to hot reservoirs is maintained, even in regimes dominated by coherent transport. We further identify the conditions that optimize TUR and MTUR violations, demonstrating that cross-correlations can enhance the joint precision of charge and heat currents near the linear-response regime.