  Quantum resources can improve communication complexity problems (CCPs) beyond
their classical constraints. One quantum approach is to share entanglement and
create correlations violating a Bell inequality, which can then assist
classical communication. A second approach is to resort solely to the
preparation, transmission and measurement of a single quantum system; in other
words quantum communication. Here, we show the advantages of the latter over
the former in high-dimensional Hilbert space. We focus on a family of CCPs,
based on facet Bell inequalities, study the advantage of high-dimensional
quantum communication, and realise such quantum communication strategies using
up to ten-dimensional systems. The experiment demonstrates, for growing
dimension, an increasing advantage over quantum strategies based on Bell
inequality violation. For sufficiently high dimensions, quantum communication
also surpasses the limitations of the post-quantum Bell correlations obeying
only locality in the macroscopic limit. Surprisingly, we find that the
advantages are tied to the use of measurements that are not rank-one
projective. We provide an experimental semi-device-independent falsification of
such measurements in Hilbert space dimension six.
