We predict an enormous order-dependent quantum enhancement of thermoelectric effects in the vicinity of higher-order interferences in the transmission spectrum of a nanoscale junction. Single-molecule junctions based on 3,3'-biphenyl and polyphenyl ether (PPE) are investigated in detail. The nonequilibrium thermodynamic efficiency and power output of a thermoelectric heat engine based on a 1,3-benzene junction are calculated using many-body theory and compared to the predictions of the figure-of-merit ZT.