Electronic structures in a hierarchical helix with inter-round couplings, i.e., a multiply twisted helix, are investigated using the spectral statistics of a quantum particle. The hierarchical structure is constructed using a procedure in which a helical chain of components is twisted, producing a doubly twisted helix, which if itself is twisted produces a triply twisted helix, and so on. We compare three types of inter-round couplings, which enable electrons to return to their original positions. When the strength of inter-round couplings is sufficiently smaller than that along a basal helix, the type of inter-round couplings can be distinguished by the behavior of spectral statistics. The multiply twisted helix of inter-round couplings through a single site is unique in the sense that its electronic structure is robust against any change in the basal helix, so is called a ‘critical helix’.
