  Phase transition of the two- and three-state quantum Potts models on the
Sierpi\'nski pyramid are studied by means of a tensor network framework, the
higher-order tensor renormalization group method. Critical values of the
transverse magnetic field and the magnetic exponent $\beta$ are evaluated.
Despite the fact that the Hausdorff dimension of the Sierpi\'nski pyramid is
exactly two $( = \log_2^{~} 4)$, the obtained critical properties show that the
effective dimension is lower than two.
