  Ideal quantum random number generators (QRNGs) can produce algorithmically
random and thus incomputable sequences, in contrast to pseudo-random number
generators. However, the verification of the presence of algorithmic randomness
and incomputability is a nontrivial task. We present the results of a search
for algorithmic randomness and incomputability in the output from two different
QRNGs, performed by applying tests based on the Solovay-Strassen test of
primality and the Chaitin-Schwartz theorem. The first QRNG uses measurements of
quantum vacuum fluctuations. The second QRNG is based on polarization
measurements on entangled single photons; for this generator, we use looped
(and thus highly compressible) strings that also allow us to assess the ability
of the tests to detect repeated bit patterns. Compared to a previous search for
algorithmic randomness, our study increases statistical power by almost 3
orders of magnitude.
