  Recent experiments on Ce$_2$Zr$_2$O$_7$ suggest that this material may host a
novel form of quantum spin ice, a three-dimensional quantum spin liquid with an
emergent photon. The Ce$^{3+}$ local moments on the pyrochlore lattice are
described by pseudospin 1/2 degrees of freedom, whose components transform as
dipolar and octupolar moments under symmetry operations. In principle, there
exist four possible quantum spin ice regimes, depending on whether the Ising
component is in the dipolar/octupolar channel, and two possible flux
configurations of the emergent gauge field. In this work, using exact
diagonalization and molecular dynamics, we investigate the equal-time and
dynamical spin structure factors in all four quantum spin ice regimes using
quantum and classical computations. Contrasting the distinct signatures of
quantum and classical results for the four possible quantum spin ice regimes
and elucidating the role of quantum fluctuations, we show that the quantum
structure factor computed for the $\pi$-flux octupolar quantum spin ice regime
is most compatible with the neutron scattering results on Ce$_2$Zr$_2$O$_7$.
