  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. In particular, if Hermiticity is
dropped, then we show that the construction is fixed uniquely up to an ordering
convention.
