  The standard formulation of quantum theory assumes a predefined notion of
time. This is a major obstacle in the search for a quantum theory of gravity,
where the causal structure of space-time is expected to be dynamical and
fundamentally probabilistic in character. Here, we propose a generalized
formulation of quantum theory without predefined time or causal structure,
building upon a recently introduced operationally time-symmetric approach to
quantum theory. The key idea is a novel isomorphism between transformations and
states which depends on the symmetry transformation of time reversal. This
allows us to express the time-symmetric formulation in a time-neutral form with
a clear physical interpretation, and ultimately drop the assumption of time. In
the resultant generalized formulation, operations are associated with regions
that can be connected in networks with no directionality assumed for the
connections, generalizing the standard circuit framework and the process matrix
framework for operations without global causal order. The possible events in a
given region are described by positive semidefinite operators on a Hilbert
space at the boundary, while the connections between regions are described by
entangled states that encode a nontrivial symmetry and could be tested in
principle. We discuss how the causal structure of space-time could be
understood as emergent from properties of the operators on the boundaries of
compact space-time regions. The framework is compatible with indefinite causal
order, timelike loops, and other acausal structures.
