  Random access codes are an intriguing class of communication tasks that
reveal an operational and quantitative difference between classical and quantum
information processing. We formulate a natural generalization of random access
codes and call them random access tests, defined for any finite collection of
measurements in an arbitrary finite dimensional general probabilistic theory.
These tests can be used to examine collective properties of collections of
measurements. We show that the violation of a classical bound in a random
access test is a signature of either measurement incompatibility or super
information storability. The polygon theories are exhaustively analyzed and a
critical difference between even and odd polygon theories is revealed.
