Given a sequence of numbers, whatever their origin, how can we assert whether they are random or not? This is an important question in many fields of computing science. Often such sequences are actually only pseudo-random, in the sense that they are produced by a deterministic algorithm, but one which is, naively speaking, chaotic enough to emulate a true uniformly distributed random sequence. In order to evaluate the quality of the randomness, statisticians have long developed a batterie of tests, one of the best known one being the so-called Diehard Test. One of the most important feature of such test suites is to verify that there are no correlations between the number in the sequence. I refer you to the Wikipedia page and the references therein if you are interested in the nitty-gritty details.
I guess, you see where I am going: has anybody thought of applying the Diehard Test, or some equivalent, to a sequence of quantum measurements. This has not been done for position as the precision of measurement is not enough and would get in the way of the actual test. But this has been done with photons, for example as reported in . I quote the authors to explain the principle of their device.
Here we present an optical Quantum Random Number Generator (QRNG), whose randomness is based on the very principles of quantum physics. The compact setup consists of a light source with stabilized intensity attenuated to the single photon level and one single photon detector. The detection events are counted during a sampling time interval $\tau_s$ and are interpreted as ’0’ for an even number of counts, whereas an odd reading corresponds to ’1’. According to fundamental laws of quantum optics the probability distribution of the number of photons in a sampling interval should follow a Poissonian distribution with mean $\mu$ for a constant intensity light source , fully analogous to radioactive sources for low $\mu$. This fact would cause a considerable bias between the num- ber of ’0’s and ’1’s in the random bit sequence. However, as we demonstrate below, dead time effects of the photomultiplier together with the read-out electronics allow to eliminate the bias even for very fast generation of random bits.
Then, the author passed their QRNG through a battery of statistical tests, including a version of Diehard, and they all passed with flying colours. Therefore in that sense, which, again, is the widely acknowledge standard of randomness, their device produces true random numbers. Such a device is particularly useful for cryptography, as a final note.
 Harald Fürst, Henning Weier, Sebastian Nauerth, Davide G. Marangon, Christian Kurtsiefer, and Harald Weinfurter. High speed optical quantum random number generation. Opt. Express, 18(12):13029–13037, Jun 2010.