  We show by embodying the Einstein equivalence principle - local Poincar\'{e}
invariance - and general covariance in quantum theory that wave-function
spreading rules out the universality of free fall, i.e. the free-fall
trajectory of a quantum (test) particle depends on its internal properties. We
provide a quantitative estimate of the free-fall non-universality in terms of
the E\"{o}tv\"{o}s parameter, which turns out to be measurable in atom
interferometry.
