  We envisage many-body systems that can be described by quantum spin-chain
Hamiltonians with a trivial separable eigenstate. For generic Hamiltonians,
such a state represents a quantum scar. We show that, typically, a
macroscopically-entangled state naturally grows after a single projective
measurement of just one spin in the trivial eigenstate; moreover, we identify a
condition under which what is growing is a "Schr\"odinger's cat state". Our
analysis does not reveal any particular requirement for the entangled state to
develop, provided that the trivial eigenstate does not minimise/maximise a
local conservation law. We study two examples explicitly: systems described by
generic Hamiltonians and a model that exhibits a $U(1)$ hidden symmetry. The
latter can be reinterpreted as a 2-leg ladder in which the interactions along
the legs are controlled by the local state on the other leg through
transistor-like building blocks.
