Epitaxial metal/semiconductor multilayers are attractive materials for a range of solid-state energy conversion devices. Applications include waste-heat-to-electrical energy conversion, hot-electron-based solar energy conversion in photocatalysis and photodiodes, optical hyperbolic metamaterials, and engineering thermal hyperconductivity. ZrN/ScN is among the first metal/semiconductor multilayer structures to also display promising thermal and electronic properties. However, for efficient thermionic transport, it is necessary to control and tune the Schottky barrier height at the metal/semiconductor interfaces, since this controls current flows across the superlattices' cross-plane directions. Sputter-deposited semiconducting ScN in ZrN/ScN multilayers contains a high concentration of n-type carriers, primarily due to oxygen impurities. This leads to a very small depletion width at metal/semiconductor interfaces, preventing thermionic transport. To overcome this challenge, the n-type carrier concentration of ScN has been reduced by Mg hole doping to similar to 1.6 x 10(18) cm(-3). In this article, we report the growth of thermally stable epitaxial ZrN/carrier-compensated Sc0.99Mg0.01N multilayers useful for thermionic emission-based devices. We present carrier concentration and transport regime calculations. Characterization of the microstructure and thermal stability was performed by combining aberration-corrected scanning transmission electron microscopy, energy-dispersive X-ray spectroscopy mapping, and atom probe tomography. The results show stoichiometric Sc0.99Mg0.01N layers with a uniform magnesium concentration, and lattice-matched ZrN/Sc0.99Mg0.01N growth with smooth and atomically sharp interfaces that are thermally stable after 48 h at 950o C. The successful demonstration of thermally stable ZrN/carrier-compensated Sc0.99Mg0.01N multilayers with a semiconductor carrier concentration of 2 x 10(18) cm(-3) is expected to enable efficient thermionic transport devices with improved properties.