Quench detection capability is essential for reliable operation and protection of superconducting magnets, coils, cables, and machinery. We propose a quench detection technique based on sensing local temperature variations in the bulk of a superconducting winding by monitoring its transient acoustic response. Our approach is primarily aimed at coils and devices built with high-temperature superconductor materials where quench detection using standard voltage-based techniques may be inefficient due to the slow velocity of quench propagation. The acoustic sensing technique is noninvasive, fast, and capable of detecting temperature variations of less than 1K in the interior of the superconductor cable stack in a 77K cryogenic environment. We show results of finite element modeling and experiments conducted on a model superconductor stack demonstrating viability of the technique for practical quench detection, discuss sensitivity limits of the technique, and its various applications. Published by AIP Publishing.