  Validity of just a few physical conditions comprising the Einstein
Equivalence Principle (EEP) suffices to ensure that gravity can be understood
as space-time geometry. EEP is therefore subject to an ongoing experimental
verification, with present day tests reaching the regime where quantum
mechanics becomes relevant. Here we show that the classical formulation of the
EEP does not apply in such a regime. The EEP requires equivalence between the
total rest mass-energy of a system, the mass-energy that constitutes its
inertia, and the mass-energy that constitutes its weight. In quantum mechanics
internal energy is given by a Hamiltonian operator describing dynamics of
internal degrees of freedom. We therefore introduce a quantum formulation of
the EEP -- equivalence between the rest, inertial and gravitational internal
energy operators. We show that the validity of the classical EEP does not imply
the validity of its quantum formulation, which thus requires an independent
experimental verification. We reanalyse some already completed experiments with
respect to the quantum EEP and discuss to which extent they allow testing its
various aspects.
