  Non-Gaussian quantum states are a known necessary resource for reaching a
quantum advantage and for violating Bell inequalities in continuous variable
systems. As one kind of manifestation of quantum correlations,
Einstein-Podolsky-Rosen (EPR) steering enables verification of shared
entanglement even when one of the subsystems is not characterized. However, how
to detect and classify such an effect for non-Gaussian states is far from being
well understood. Here, we present an efficient non-Gaussian steering criterion
based on the high-order observables and conduct a systematic investigation into
the hierarchy of non-Gaussian steering criteria. Moreover, we apply our
criterion to three experimentally-relevant non-Gaussian states under realistic
conditions and, in particular, propose a feasible scheme to create
multi-component cat states with tunable size by performing a suitable
high-order quadrature measurement on the steering party. Our work reveals the
fundamental characteristics of non-Gaussianity and quantum correlations, and
offers new insights to explore their applications in quantum information
processing.
