  The generation of nonuniform quadrupole states plays a crucial role in
understanding various fascinating phenomena observed in the advancement of
several research areas, e.g., multiferroic compounds, nonmagnetic
superconductors, etc. In this work, we investigate the ground-state phase
diagram of a generalized spin-3/2 bilinear-biquadratic-bicubic Heisenberg chain
in the representation of multipolar operators. By numerical simulations with
the large-scale density-matrix renormalization group (DMRG) method, we
successfully identify a tetramerization phase and a stripe-Q phase. These
phases are characterized by the emergence of nonuniform quadrupole orders
resulting from the spontaneous breaking of translation symmetry. In particular,
tetramerization phase refers to the quadrupole operators take a four-cycle,
while the stripe-Q phase represents a striped pattern in quadrupole operators.
Additionally, we demonstrate the presence of a Wess-Zumino-Witten (WZW) model
with level k = 1 at the transition point between the dimerized (DM) phase and
the Luttinger liquid (LL) phase, based on strong numerical findings.
