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

Random access test as an identifier of nonclassicality