  Quasicrystals are fascinating and important because of their unconventional
atomic arrangements, which challenge traditional notions of crystalline
structures. Unlike regular crystals, they lack translational symmetry and
generate unique mechanical, thermal, and electrical properties, holding promise
for numerous applications. In order to probe the electronic properties of
quasicrystals, tools beyond linear response transport measurements are needed,
since all spectral regions can be affected by the non-periodic geometry. Here
we show that high-harmonic spectroscopy offers an advanced avenue for this
goal. Focusing on the quasiperiodic 1D Aubry-Andr\'e-Harper (AAH) model, we
leverage high-harmonic spectroscopy to delve into their intricate
characteristics: By carefully analyzing emitted harmonic intensities, we
extract the multifractal spectrum -- an essential indicator of the spatial
distribution of electronic states in quasicrystals. Additionally, we address
the detection of mobility edges, vital energy thresholds that demarcate
localized and extended eigenstates within generalized AAH models. The precise
identification of these mobility edges sheds light on the metal-insulator
transition and the behavior of electronic states near these boundaries. Merging
high-harmonic spectroscopy with the AAH model provides a powerful framework for
understanding the interplay between localization and extended states in
quasicrystals for an extremely wide energy range not captured within linear
response studies, thereby offering valuable insights for guiding future
experimental investigations.
