  Quantum critical points in quasiperiodic magnets can realize new universality
classes, with critical properties distinct from those of clean or disordered
systems. Here, we study quantum phase transitions separating ferromagnetic and
paramagnetic phases in the quasiperiodic $q$-state Potts model in $2+1d$. Using
a controlled real-space renormalization group approach, we find that the
critical behavior is largely independent of $q$, and is controlled by an
infinite-quasiperiodicity fixed point. The correlation length exponent is found
to be $\nu=1$, saturating a modified version of the Harris-Luck criterion.
