  The experimental violation of a Bell inequality implies that at least one of
a set of assumptions fails in nature. However, existing tests are inconclusive
about which of the assumptions is the one that fails. Here, we show that there
are quantum correlations that cannot be simulated with hidden variables that
allow the slightest free will (or, equivalently, that limit, even minimally,
retrocausal influences) or restrict, even minimally, actions at a distance.
This result goes beyond Bell's theorem and demolishes the arguably most
attractive motivation for considering hidden-variable theories with measurement
dependence or actions at distance, namely, that simulating quantum correlations
typically requires a small amount of these resources. We show that there is a
feasible experiment that can discard any hidden-variable theory allowing for
arbitrarily small free will and having arbitrarily small limitations to actions
at a distance. The experiment involves two observers, each of them choosing
between two measurements with $2^N$ outcomes. The larger $N$ for which a
specific Bell-like inequality is violated, the larger the set of excluded
hidden-variable theories. In the limit of $N$ tending to infinity, the only
alternatives to the absence of hidden variables are complete superdeterminism
or complete parameter dependence. We also explore the implications of this
result for quantum information.
