  There is a stark tension among different formulations of quantum theory in
that some are fundamentally time-symmetric and others are radically
time-asymmetric. This tension is crisply captured when thinking of physical
theories as theories of processes. We review process theories and their
diagrammatic representation, and show how quantum theory can be described in
this language. The tension between time-symmetry and time-asymmetry is then
captured by the tension between two of the key structures that are used in this
framework. On the one hand, the symmetry is captured by a dagger structure,
which is represented by a reflection of diagrams. On the other hand, the
asymmetry is captured by a condition involving discarding which, ultimately, is
responsible for the theory being compatible with relativistic causality.
  Next we consider three different ways in which we this tension can be
resolved. The first of these is closely related to recent work of Lucien Hardy,
where the tension is resolved by adding in a time reversed version of
discarding together with a suitable consistency condition. The second is, to
our knowledge, a new approach. Here the tension is resolved by adding in new
systems which propagate backwards in time, and imposing a consistency condition
to avoid running into well known time-travel paradoxes. The final approach that
we explore is closely related to work of Oreshkov and Cerf, where the tension
is resolved by removing the constraint associated with discarding. We show two
equivalent ways in which this can be done whilst ensuring that the resulting
theory still makes sensible operational predictions.
