  Topology ultimately unveils the roots of the perfect quantization observed in
complex systems. The 2D quantum Hall effect is the celebrated archetype.
Remarkably, topology can manifest itself even in higher-dimensional spaces in
which control parameters play the role of extra, synthetic dimensions. However,
so far, a very limited number of implementations of higher-dimensional
topological systems have been proposed, a notable example being the so-called
4D quantum Hall effect. Here we show that mesoscopic superconducting systems
can implement higher-dimensional topology and represent a formidable platform
to study a quantum system with a purely nontrivial second Chern number. We
demonstrate that the integrated absorption intensity in designed microwave
spectroscopy is quantized and the integer is directly related to the second
Chern number. Finally, we show that these systems also admit a non-Abelian
Berry phase. Hence, they also realize an enlightening paradigm of topological
non-Abelian systems in higher dimensions.
