  A class of quantum resource theories, based on non-convex star-shape sets,
presented in this work captures the key quantum properties that cannot be
studied by standard convex theories. We provide operational interpretations for
a resource of this class and demonstrate its advantage to improve performance
of correlated quantum discrimination tasks and testing of quantum combs.
Proposed techniques provide useful tools to describe quantum discord, total
correlations in composite quantum systems and to estimate the degree of
non-Markovianity of an analyzed quantum dynamics. Other applications include
the problem of unistochasticity of a given bistochastic matrix, with relevance
for quantization of classical dynamics and studies of violation of CP-symmetry
in high energy physics. In all these cases, the non-linear witnesses introduced
here outperform the standard linear witnesses. Importance of our findings for
quantum information theory is also emphasized.
