  Higher-order topological insulators are an intriguing new family of
topological states that host lower-dimensional boundary states. Concurrently,
quasiperiodic systems have garnered significant interest due to their complex
localization and topological properties. In this work we study the impact of
quasiperiodic modulations on the paradigmatic Benalcazar-Bernevig-Hughes model,
which hosts topological insulating phases with zero-energy corner modes. We
find that the topological properties are not only robust to the quasiperiodic
modulation, but can even be enriched. In particular, we unveil the first
instance of a quasiperiodic induced second-order topological insulating phase.
Furthermore, in contrast with disorder, we find that quasiperiodic modulations
can induce multiple reentrant topological transitions, showing an intricate
sequence of localization properties. Our results open a promising avenue for
exploring the rich interplay between higher-order topology and
quasiperiodicity.
