  Multipolar Kondo systems offer unprecedented opportunities for designing
astonishing quantum phases and functionalities beyond spin-only descriptions. A
model material platform of this kind is the cubic heavy-fermion system
Pr$Tr_{2}$Al$_{20}$ ($Tr=$ Ti, V), which hosts a nonmagnetic
crystal-electric-field (CEF) ground state and substantial Kondo entanglement of
the local quadrupolar and octopolar moments with the conduction electron sea.
Here, we explore magnetoresistance (MR) and Hall effect of PrTi$_{2}$Al$_{20}$
that develops ferroquadrupolar (FQ) order below $T_{Q} \sim 2$ K and compare
its behavior with that of the non-4$f$ analog, LaTi$_{2}$Al$_{20}$. In the FQ
ordered phase, PrTi$_{2}$Al$_{20}$ displays extremely large magnetoresistance
(XMR) of $\sim 10^{3}\%$. The unsaturated, quasi-linear field ($B$) dependence
of the XMR violates the conventional Kohler's scaling and defies description
based on carrier compensation alone. By comparing the MR and the Hall effect
observed in PrTi$_{2}$Al$_{20}$ and LaTi$_{2}$Al$_{20}$, we conclude that the
open-orbit topology on the electron-type Fermi surface (FS) sheet is key for
the observed XMR. The low-temperature MR and the Hall resistivity in
PrTi$_{2}$Al$_{20}$ display pronounced anisotropy in the [111] and [001]
magnetic fields, which is absent in LaTi$_{2}$Al$_{20}$, suggesting that the
transport anisotropy ties in with the anisotropic magnetic-field response of
the quadrupolar order parameter.
