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Strong quantum zero-point motion (ZPM) of light nuclei and other particles is a crucial aspect of many state-of-the-art quantum materials. However, it has only recently begun to be explored from an $\textit{ab initio}$ perspective, through several competing approximations. Here we develop a unified description of muon and light nucleus ZPM and establish the regimes of anharmonicity and positional quantum entanglement where different approximation schemes apply. Via density functional theory and path-integral molecular dynamics simulations we demonstrate that in solid nitrogen, $α\unicode{x2013}$N$_2$, muon ZPM is both strongly anharmonic and many-body in character, with the muon forming an extended electric-dipole polaron around a central, quantum-entangled [N$_2\unicode{x2013}μ\unicode{x2013}$N$_2$]$^+$ complex. By combining this quantitative description of quantum muon ZPM with precision muon quadrupolar level-crossing resonance experiments, we independently determine the static $^{14}$N nuclear quadrupolar coupling constant of pristine $α\unicode{x2013}$N$_2$ to be $-5.36(2)$ MHz, a significant improvement in accuracy over the previously-accepted value of $-5.39(5)$ MHz, and a validation of our unified description of light-particle ZPM.