If an observer measures a system once, it projected to some eigenstate A. He save this information, goes around the CTC, measure the system again. Then in order to ensure self consistency, the outcome will always be A?
Put it in another way, suppose we let the measurement thing to be part of the evolution that happens to evolve the quantum state arbitrarily close to an eigenstate at some time $t'\in [0,T]$, then we could wrote:
$$\lvert\psi(t)\rangle=U(t,t_0)\lvert\psi (t_0)\rangle$$
and then we could demand that $\lvert\psi (0)\rangle=\lvert\psi (T)\rangle$ as the boundary condition of …