Quantized Electric Multipole Insulators: Summary This paper extends the Berry-phase formulation of electric polarization in crystals to higher electric multipole moments. The authors determine the conditions under which quadrupole and octupole moments are topologically quantized electromagnetic observables. These systems exhibit gapped boundaries that are themselves lower-dimensional topological phases and manifest topologically protected corner states carrying fractional charge. The authors introduce a new paradigm using nested Wilson loops to generate topological invariants, and propose three experimental implementations of this behavior. The paper discusses the quantization of physical phenomena such as charge polarization, Hall conductance, and magneto-electric polarizability in topological insulators. These quantities are expressed as quantized topological invariants in terms of the Berry phase vector potential. The authors then generalize these results to higher electric multipole moments, showing that they are also topologically quantized and protected by certain crystalline symmetries. The paper presents a model with a quantized quadrupole moment, showing that it leads to quantized boundary polarizations and corner charges. The model is analyzed using Wilson loops and nested Wilson loops, which reveal the topological nature of the quadrupole moment. The authors also discuss the experimental realization of this model in cold-atom lattice systems and photonic crystals. The paper also presents a model with a quantized octupole moment, showing that it leads to quantized surface quadrupoles and hinge polarizations. The authors analyze the topological nature of the octupole moment using Wilson loops and nested Wilson loops, and discuss the experimental realization of this model in photonic crystals. The paper concludes by summarizing the key findings: the quantization of higher electric multipole moments in crystals, the topological protection of these moments by certain symmetries, and the experimental realization of these phenomena in cold-atom and photonic systems. The authors also discuss the implications of these findings for the broader field of topological insulators and the potential for future research.Quantized Electric Multipole Insulators: Summary This paper extends the Berry-phase formulation of electric polarization in crystals to higher electric multipole moments. The authors determine the conditions under which quadrupole and octupole moments are topologically quantized electromagnetic observables. These systems exhibit gapped boundaries that are themselves lower-dimensional topological phases and manifest topologically protected corner states carrying fractional charge. The authors introduce a new paradigm using nested Wilson loops to generate topological invariants, and propose three experimental implementations of this behavior. The paper discusses the quantization of physical phenomena such as charge polarization, Hall conductance, and magneto-electric polarizability in topological insulators. These quantities are expressed as quantized topological invariants in terms of the Berry phase vector potential. The authors then generalize these results to higher electric multipole moments, showing that they are also topologically quantized and protected by certain crystalline symmetries. The paper presents a model with a quantized quadrupole moment, showing that it leads to quantized boundary polarizations and corner charges. The model is analyzed using Wilson loops and nested Wilson loops, which reveal the topological nature of the quadrupole moment. The authors also discuss the experimental realization of this model in cold-atom lattice systems and photonic crystals. The paper also presents a model with a quantized octupole moment, showing that it leads to quantized surface quadrupoles and hinge polarizations. The authors analyze the topological nature of the octupole moment using Wilson loops and nested Wilson loops, and discuss the experimental realization of this model in photonic crystals. The paper concludes by summarizing the key findings: the quantization of higher electric multipole moments in crystals, the topological protection of these moments by certain symmetries, and the experimental realization of these phenomena in cold-atom and photonic systems. The authors also discuss the implications of these findings for the broader field of topological insulators and the potential for future research.