  We investigate the complexity cost of demonstrating the key types of
nonclassical correlations --- Bell inequality violation, EPR-steering, and
entanglement --- with independent agents, theoretically and in a photonic
experiment. We show that the complexity cost exhibits a hierarchy among these
three tasks, mirroring the recently-discovered hierarchy for how robust they
are to noise. For Bell inequality violations, the simplest test is the
well-known CHSH test, but for EPR-steering and entanglement the tests that
involve the fewest number of detection patterns require non-projective
measurements. The simplest EPR-steering requires a choice of projective
measurement for one agent and a single non-projective measurement for the
other, while the simplest entanglement test uses just a single non-projective
measurement for each agent. In both of these cases, we derive our inequalities
using the concept of circular 2-designs. This leads to the interesting feature
that in our photonic demonstrations, the correlation of interest is independent
of the angle between the linear polarizers used by the two parties, which thus
require no alignment.
