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.
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