Charge transport of topological semimetals is the focus of intensive investigations because of their nontrivial band topology. Heat transport of these materials, on the other hand, is largely unexplored and remains elusive. Here, we report on an observation of unprecedented, giant magnetic quantum oscillations of thermal conductivity in the prototypical Weyl semimetal TaAs. The oscillations are antiphase with the quantum oscillating electronic density of states of a Weyl pocket, and their amplitudes amount to 2 orders of magnitude of the estimation based on the Wiedemann-Franz law. Our analyses show that all the conventional heat-transport mechanisms through diffusions of propagating electrons, phonons, and electron-hole bipolar excitations are far inadequate to account for these phenomena. Taking further experimental facts that the parallel field configuration favors much-higher magnetothermal conductivity, we propose that the newly proposed chiral zero sound provides a reasonable explanation to these exotic phenomena. More work focusing on other topological semimetals along the same line is badly called for.