  Can collective quantum effects make a difference in a meaningful
thermodynamic operation? Focusing on energy storage and batteries, we
demonstrate that quantum mechanics can lead to an enhancement in the amount of
work deposited per unit time, i.e., the charging power, when $N$ batteries are
charged collectively. We first derive analytic upper bounds for the collective
\emph{quantum advantage} in charging power for two choices of constraints on
the charging Hamiltonian. We then highlight the importance of entanglement by
proving that the quantum advantage vanishes when the collective state of the
batteries is restricted to be in the separable ball. Finally, we provide an
upper bound to the achievable quantum advantage when the interaction order is
restricted, i.e., at most $k$ batteries are interacting. Our result is a
fundamental limit on the advantage offered by quantum technologies over their
classical counterparts as far as energy deposition is concerned.
