Magneto-quantum oscillation experiments in high-temperature superconductors show a strong thermally induced suppression of the oscillation amplitude approaching the critical dopings [B. J. Ramshaw , Science 348, 317 (2014)SCIEAS0036-807510.1126/science.aaa4990; H. Shishido , Phys. Rev. Lett. 104, 057008 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.057008; P. Walmsley , Phys. Rev. Lett. 110, 257002 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.257002]—in support of a quantum-critical origin of their phase diagrams. We suggest that, in addition to a thermodynamic mass enhancement, these experiments may directly indicate the increasing role of quantum fluctuations that suppress the quantum oscillation amplitude through inelastic scattering. We show that the traditional theoretical approaches beyond Lifshitz-Kosevich to calculate the oscillation amplitude in correlated metals result in a contradiction with the third law of thermodynamics and suggest a way to rectify this problem.