  Electronic materials harbor a plethora of exotic quantum phases, ranging from
unconventional superconductors to non-Fermi liquids, and, more recently,
topological phases of matter. While these quantum phases in integer dimensions
are well characterized by now, their presence in fractional dimensions remains
vastly unexplored. Here, we theoretically show that a special class of
crystalline phases, namely, higher-order topological phases that via an
extended bulk-boundary correspondence feature robust gapless modes on lower
dimensional boundaries, such as corners and hinges, can be found on a
representative family of fractional materials: \emph{quantum fractals}. To
anchor this general proposal, we demonstrate realizations of second-order
topological insulators and superconductors, supporting charged and neutral
Majorana corner modes, on planar Sierpi\'{n}ski carpet and triangle fractals,
respectively. These predictions can be experimentally tested on designer
electronic fractal materials, as well as on various highly tunable metamaterial
platforms, such as photonic and acoustic lattices.
