  We propose {\it measurement-producing hierarchy} emerging among correlated
states by sequential subsystem projective measurements. We start from
symmetry-protected-topological (SPT) cluster states with a large symmetry and
apply sequential subsystem projective measurements to them and find that
generalized cluster SPT states with a reduced symmetry appear in the subsystem
of the unmeasured sites. That prescription finally produces
Greenberger-Home-Zeilinger states with long-range order in the subsystem
composed of periodic unmeasured sites of the original lattice. The
symmetry-reduction hierarchical structure from a general large symmetric SPT
cluster state is clearly captured by the measurement update flow in the
efficient algorithm of stabilizer formalism. This approach is useful not only
for the analytical search for the measured state but also for numerical
simulation with a large system size. We also numerically verify the
symmetry-reduction hierarchy by sequential subsystem projective measurements
applied to large systems and large symmetric cluster SPT states.
