  When an experimentalist measures a time series of qubits, the outcomes
generate a classical stochastic process. We show that measurement induces high
complexity in these processes in two specific senses: they are inherently
unpredictable (positive Shannon entropy rate) and they require an infinite
number of features for optimal prediction (divergent statistical complexity).
We identify nonunifilarity as the mechanism underlying the resulting
complexities and examine the influence that measurement choice has on the
randomness and structure of measured qubit processes. We introduce new
quantitative measures of this complexity and provide efficient algorithms for
their estimation.
