de Sitter space, extremal surfaces, and time entanglement

We refine previous investigations on de Sitter space and extremal surfaces anchored at the future boundary $I^+$. Since such do not return, they require extra data or conditions in past (interior). In entirely Lorentzian spacetime, this leads to future-past timelike stretching between $I^\pm$. Apart from an overall $-i$ factor (relative spacelike $AdS$) their areas are real positive. With a no-boundary type condition, top half of these joins with part hemisphere giving complex-valued area. Motivated by these, we describe two aspects "time-entanglement" simple toy models quantum mechanics. One is based thermofield double state entangling separated states, which positive structures. Another time evolution operator reduced transition amplitudes, entropy.