  In a recent work [Phys. Rev. Lett. 116, 240401 (2016)], a framework known by
the name of "assemblage moment matrices" (AMMs) has been introduced for the
device-independent quantification of quantum steerability and measurement
incompatibility. In other words, even with no assumption made on the
preparation device nor the measurement devices, one can make use of this
framework to certify, directly from the observed data, the aforementioned
quantum features. Here, we further explore the framework of AMM and provide
improved device-independent bounds on the generalized robustness of
entanglement, the incompatibility robustness and the incompatibility weight. We
compare the tightness of our device-independent bounds against those obtained
from other approaches. Along the way, we also provide an analytic form for the
generalized robustness of entanglement for an arbitrary two-qudit isotropic
state. When considering a Bell-type experiment in a tri- or more-partite
scenario, we further show that the framework of AMM provides a natural way to
characterize a superset to the set of quantum correlations, namely, one which
also allows post-quantum steering.
