  Characterizing quantum correlations in terms of information-theoretic
principles is a popular chapter of quantum foundations. Traditionally, the
principles adopted for this scope have been expressed in terms of conditional
probability distributions, specifying the probability that a black box produces
a certain output upon receiving a certain input. This framework is known as
"device-independent". Another major chapter of quantum foundations is the
information-theoretic characterization of quantum theory, with its sets of
states and measurements, and with its allowed dynamics. The different
frameworks adopted for this scope are known under the umbrella term "general
probabilistic theories". With only a few exceptions, the two programmes on
characterizing quantum correlations and characterizing quantum theory have so
far proceeded on separate tracks, each one developing its own methods and its
own agenda. This paper aims at bridging the gap, by comparing the two
frameworks and illustrating how the two programmes can benefit each other.
