  While we have intuitive notions of structure and complexity, the
formalization of this intuition is non-trivial. The statistical complexity is a
popular candidate. It is based on the idea that the complexity of a process can
be quantified by the complexity of its simplest mathematical model - the model
that requires the least past information for optimal future prediction. Here we
review how such models, known as $\epsilon$-machines can be further simplified
through quantum logic, and explore the resulting consequences for understanding
complexity. In particular, we propose a new measure of complexity based on
quantum $\epsilon$-machines. We apply this to a simple system undergoing
constant thermalization. The resulting quantum measure of complexity aligns
more closely with our intuition of how complexity should behave.
