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

High Energy Physics - Theory