Can a quantum state over time resemble a quantum state at a single time?

Understand

Standard quantum theory represents a composite system at a given time by a joint state, but it does not prescribe a joint state for a composite of systems at different times.

• If a more even-handed treatment of space and time is possible, then such a joint state should be definable, and one might expect it to satisfy the following five conditions: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time-steps is associative.
• We show that no construction satisfies all these requirements.
• If an even-handed treatment of space and time is possible, therefore, one or more axioms must be dropped.