  Wave-particle duality, superposition and entanglement are among the most
counterintuitive features of quantum theory. Their clash with our classical
expectations motivated hidden-variable (HV) theories. With the emergence of
quantum technologies we can test experimentally the predictions of quantum
theory {\em versus} HV theories and put strong restrictions on their key
assumptions. Here we study an entanglement-assisted version of the quantum
delayed-choice experiment and show that the extension of HV to the controlling
devices only exacerbates the contradiction. We compare HV theories that satisfy
the conditions of objectivity (a property of photons being either particles or
waves, but not both), determinism, and local independence of hidden variables
with quantum mechanics. Any two of the above conditions are compatible with it.
The conflict becomes manifest when all three conditions are imposed and
persists for any non-zero value of entanglement. We propose an experiment to
test our conclusions.
