  Recently there have been a number of experiments on Ce$_2$Zr$_2$O$_7$ and
Ce$_2$Sn$_2$O$_7$, suggesting that these materials host a three-dimensional
quantum spin liquid with emergent photons and fractionalized spinon
excitations. However, the interpretation of the data to determine the precise
nature of the quantum spin liquids is still under debate. The Kramers doublet
in Ce$^{3+}$ local moment offers unusual pseudo-spin degrees of freedom as the
$x$ and $z$ components transform as a dipole and $y$ component as an octupole.
This leads to a variety of possible quantum spin liquid (or quantum spin ice)
phases on the pyrochlore lattice of Ce$^{3+}$ moments. In this work, we
theoretically propose that magnetostriction would be able to distinguish the
dipolar (D-QSI) and octupolar (O-QSI) quantum spin ice, where the dipolar or
octupolar components possess the respective spin ice correlations. We show that
the magnetostriction in various configurations can be used as a selection rule
to differentiate not only D-QSI and O-QSI, but also a number of competing
broken symmetry states.
