  We develop a framework for characterizing quantum temporal correlations in a
general temporal scenario, in which an initial quantum state is measured, sent
through a quantum channel, and finally measured again. This framework does not
make any assumptions on the system nor on the measurements, namely, it is
device-independent. It is versatile enough, however, to allow for the addition
of further constraints in a semi-device-independent setting. Our framework
serves as a natural tool for quantum certification in a temporal scenario when
the quantum devices involved are uncharacterized or partially characterized. It
can hence also be used for characterizing quantum temporal correlations when
one assumes an additional constraint of no-signalling in time, there are upper
bounds on the involved systems' dimensions, rank constraints -- for which we
prove genuine quantum separations over local hidden variable models -- or
further linear constraints. We present a number of applications, including
bounding the maximal violation of temporal Bell inequalities, quantifying
temporal steerability, bounding the maximum successful probability in quantum
randomness access codes.
