  We report two fundamental experiments on three-level quantum systems
(qutrits). The first one tests the simplest task for which quantum mechanics
provides an advantage with respect to classical physics. The quantum advantage
is certified by the violation of Wright's inequality, the simplest classical
inequality violated by quantum mechanics. In the second experiment, we obtain
contextual correlations by sequentially measuring pairs of compatible
observables on a qutrit, and show the violation of Klyachko et al.'s
inequality, the most fundamental noncontextuality inequality violated by
qutrits. Our experiment tests exactly Klyachko et al.'s inequality, uses the
same measurement procedure for each observable in every context, and shows that
the violation does not depend on the order of the measurements.
