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Time interfaces consist of abrupt and spatially uniform changes in the optical properties of a medium. Their periodic occurrence forms photonic time crystals, which offer opportunities to tailor classical and quantum light-matter interactions. Here we explore one-dimensional photonic time quasicrystals, formed when time interfaces occur in a quasiperiodic fashion featuring long-range order. We unveil the emergence of topological phases and Hofstadter butterfly spectra in these systems, and demonstrate that their temporal response emulates the localization of topological edge states, enabling the temporal analog of topological Thouless pumping. Our findings open avenues for topological photonics leveraging time as a synthetic dimension, with functionalities that go beyond their spatial counterparts.
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