  Recently, the studies of topological corner states (TCSs) are extended from
crystals to quasicrystals, which are referred to as higher-order topological
quasicrystalline insulators (HOTQIs). However, the TCSs of complete
quasi-periodic structure in photonic systems have yet to be demonstrated.
Moreover, there is only one TCS in each corner region in higher-order
topological insulators (HOTIs). Increasing the number of TCS is expected to
increase application potential of TCSs. In this work, HOTQIs in photonic
systems are experimentally observed. It is found that HOTQIs possess TCS
arrays, and each TCS array contains several TCSs. Furthermore, the universal
theoretical framework of the multimer analysis method is improved, and the
difference in the average charge density is proposed as a real-space
topological index. These results will open up new ideas for investigating
highly integrated, multi-region localized TCSs and are expected to provide new
ways to explore topological phenomena and the applications of photonic
quasicrystals.
