Polyadic Higher-Order Fixpoint Logic (PHFL) is a modal fixpoint logic obtained as the merger of Higher-Order Fixpoint Logic (HFL) and the Polyadic μ-Calculus. Polyadicity enables formulas to make assertions about tuples of states rather than states only. Like HFL, PHFL has the ability to formalise properties using higher-order functions. We consider PHFL in the setting of descriptive complexity theory: its fragment using no functions of higher-order is exactly the Polyadic μ-Calculus, and it is known from Otto’s Theorem that it captures the bisimulation-invariant fragment of PTIME. We extend this and give capturing results for the bisimulation-invariant fragments of EXPTIME, PSPACE, and NLOGSPACE.