  Coherence and entanglement are fundamental properties of quantum systems,
promising to power the near future quantum computers, sensors and simulators.
Yet, their experimental detection is challenging, usually requiring full
reconstruction of the system state. We show that one can extract quantitative
bounds to the relative entropy of coherence and the coherent information,
coherence and entanglement quantifiers respectively, by a limited number of
purity measurements. The scheme is readily implementable with current
technology to verify quantum computations in large scale registers, without
carrying out expensive state tomography.
