  Process matrices are a framework to model causal relations in the absence of
a well-defined acyclic causal order. The framework is very general and does not
even assume the existence of a background spacetime. As a result, it is an open
question how the framework should be interpreted physically and how and even if
composition can be defined. On the other hand, so-called causal boxes define a
framework that allows for arbitrary composition. In this work, we treat quantum
circuits with quantum control of causal order (QC-QC), a subset of process
matrices, which can be interpreted as generalized quantum circuits, and process
box, a subset of causal boxes, which can be interpreted as processes. We
analyze their state spaces and define a notion of operational equivalence
between QC-QCs and process boxes based on this analysis. We then explicitly
construct for each QC-QC an operationally equivalent process box. This allows
us to define composition of QC-QCs in terms of composition of causal boxes
which is well-defined. We further show that process boxes admit a unitary
extension and conjecture that the background spacetime can be assumed to have a
specific simple form. Based on this conjecture, we construct an operationally
equivalent QC-QC for each process box. Our results indicate that the only class
of processes that can be physically implemented in a fixed background spacetime
are those that can be interpreted as quantum circuits with quantum controlled
superpositions of orders. Further, they also reveal that the composability
issue can be resolved by embedding processes in a spacetime structure. This in
turn sheds light on the connection between physical realizability in a
spacetime and composability.
