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We envisage many-body systems that can be described by quantum spin-chainHamiltonians with a trivial separable eigenstate. For generic Hamiltonians,such a state represents a quantum scar. We show that, typically, amacroscopically-entangled state naturally grows after a single projectivemeasurement of just one spin in the trivial eigenstate; moreover, we identify acondition under which what is growing is a "Schr\"odinger's cat state". Ouranalysis does not reveal any particular requirement for the entangled state todevelop, provided that the trivial eigenstate does not minimise/maximise alocal conservation law. We study two examples explicitly: systems described bygeneric Hamiltonians and a model that exhibits a $U(1)$ hidden symmetry. Thelatter can be reinterpreted as a 2-leg ladder in which the interactions alongthe legs are controlled by the local state on the other leg throughtransistor-like building blocks.

Growing Schr\"odinger's cat states by local unitary time evolution of product states