  Recently extended from the modern theory of electric polarization, quantized
multipole topological insulators (QMTIs) describe higher-order multipole
moments, lying in nested Wilson loops, which are inherently quantized by
lattice symmetries. Overlooked in the past, QMTIs reveal new types of gapped
boundaries, which themselves represent lower-dimensional topological phases and
host topologically protected zero-dimensional (0D) corner states. Inspired by
these pioneering theoretical predictions, tremendous efforts have been devoted
to the experimental observation of topological quantized quadrupole phase in a
variety of two dimensional (2D) metamaterials. However, due to stringent
requirements of anti-commuting reflection symmetries in crystals, it has been
challenging to achieve higher-order quantized multipole moments, such as
octupole moments, in a realistic three-dimensional (3D) structure. Here, we
overcome these challenges, and experimentally realize the acoustic analogue of
a quantized octupole topological insulator (QOTIs) using negatively coupled
resonators. We confirm by first-principle studies that our design possesses a
quantized octupole topological phase, and experimentally demonstrate
spectroscopic evidence of a topological hierarchy of states in our
metamaterial, observing 3rd order corner states, 2nd order hinge states and 1st
order surface states. Furthermore, we reveal topological phase transitions from
higher- to lower-order multipole moments in altered designs of acoustic TIs.
Our work offers a new pathway to explore higher-order topological states
(HOTSs) in 3D classical platforms.
