Is the randomness generated by quantum phenomena measurably better than the randomness generated by the best computational algorithms? I am aware that quantum phenomena can be used to generate random sequences of numbers. Are such sequences measurably better than the random sequences generated by the best computational algorithms?  If so, how much better, and according to what measurements? I'm interested in measurements that can be computed in reasonable time and energy, not theoretical measurements.  If you refer to a measurement, please give an estimate of the amount of time and energy the measurement requires to produce a result, if possible.
 A: No: the output of current cryptographic PRGs is not distinguishable from true randomness by any known algorithm (in a practical amount of time).
For instance, there's no known way of telling whether a stream of bits was produced by the ChaCha20 stream cipher that's substantially faster than evaluating the cipher about $2^{128}$ times on a quantum computer or $2^{256}$ times on a classical computer, which is thoroughly impractical with current or even conceivable future computing technology. It's hard to put any specific cost on it, but, e.g., if your quantum computer could compute one cipher block per picosecond, it would take around $10^{19}$ years to check all keys.
A: Any probability distribution can be generated classically, and similarly, you can use classical physics to generate any bipartite probability distribution.
Still, quantum phenomena can result in types of correlations that are classically impossible (and yes, that can be measured experimentally without significant issues).
But one needs to be careful in understand exactly what sorts of "correlations" we are talking about here.
It's not that with QM you can get correlations between measurement outcomes that are "non-classical", as that wouldn't really make much sense at all.
What you can get, however, are "correlations" between correlations observed in the different measurement bases, that are nonclassical.
In other words, two quantum states can be correlated in such a way that the correlations between measurement outcomes resulting from different measurement choices are impossible to obtain via any form of classical correlations.
