Interaction induced topological states remain one of the most fascinating phenomena in condensed matter physics. The exciton condensate has recently sparked renewed interest due to the discovery of new candidate materials and its driving force to realize exotic topological states. In this work, we explore the exciton order induced high-order topology in the bilayer quantum spin Hall insulators and find that the topological excitonic corner states can be realized by tuning the gate and magnetic field. When an in-plane Zeeman field is applied to the system, two or four excitonic boundary-obstructed corner states emerge in the bilayer system for distinct possible $s$-wave excitonic pairings. Besides, we also find a two-dimensional excitonic Weyl nodal phase, which supports flat band edge states connecting the bulk Weyl nodes.

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• Cellular bilayers
• Excitons
• Spin Hall effect
• Topological phases of matter