  High-dimensional quantum entanglement can give rise to stronger forms of
nonlocal correlations compared to qubit systems, offering significant
advantages for quantum information processing. Certifying these stronger
correlations, however, remains an important challenge, in particular in an
experimental setting. Here we theoretically formalise and experimentally
demonstrate a notion of genuine high-dimensional quantum steering. We show that
high-dimensional entanglement, as quantified by the Schmidt number, can lead to
a stronger form of steering, provably impossible to obtain via entanglement in
lower dimensions. Exploiting the connection between steering and
incompatibility of quantum measurements, we derive simple two-setting steering
inequalities, the violation of which guarantees the presence of genuine
high-dimensional steering, and hence certifies a lower bound on the Schmidt
number in a one-sided device-independent setting. We report the experimental
violation of these inequalities using macro-pixel photon-pair entanglement
certifying genuine high-dimensional steering. In particular, using an entangled
state in dimension $d=31$, our data certifies a minimum Schmidt number of
$n=15$.
