  The conflict between classical and quantum physics can be identified through
a series of yes-no tests on quantum systems, without it being necessary that
these systems be in special quantum states. Kochen-Specker (KS) sets of yes-no
tests have this property and provide a quantum-versus-classical advantage that
is free of the initialization problem that affects some quantum computers.
Here, we report the first experimental implementation of a complete KS set that
consists of 18 yes-no tests on four-dimensional quantum systems and show how to
use the KS set to obtain a state-independent quantum advantage. We first
demonstrate the unique power of this KS set for solving a task while avoiding
the problem of state initialization. Such a demonstration is done by showing
that, for 28 different quantum states encoded in the orbital-angular-momentum
and polarization degrees of freedom of single photons, the KS set provides an
impossible-to-beat solution. In a second experiment, we generate maximally
contextual quantum correlations by performing compatible sequential
measurements of the polarization and path of single photons. In this case,
state independence is demonstrated for 15 different initial states. Maximum
contextuality and state independence follow from the fact that the sequences of
measurements project any initial quantum state onto one of the KS set's
eigenstates. Our results show that KS sets can be used for quantum-information
processing and quantum computation and pave the way for future developments.
