  The advantages of quantum random number generators (QRNGs) over pseudo-random
number generators (PRNGs) are normally attributed to the nature of quantum
measurements. This is often seen as implying the superiority of the sequences
of bits themselves generated by QRNGs, despite the absence of empirical tests
supporting this. Nonetheless, one may expect sequences of bits generated by
QRNGs to have properties that pseudo-random sequences do not; indeed,
pseudo-random sequences are necessarily computable, a highly nontypical
property of sequences.
  In this paper, we discuss the differences between QRNGs and PRNGs and the
challenges involved in certifying the quality of QRNGs theoretically and
testing their output experimentally. While QRNGs are often tested with standard
suites of statistical tests, such tests are designed for PRNGs and only verify
statistical properties of a QRNG, but are insensitive to many supposed
advantages of QRNGs. We discuss the ability to test the incomputability and
algorithmic complexity of QRNGs. While such properties cannot be directly
verified with certainty, we show how one can construct indirect tests that may
provide evidence for the incomputability of QRNGs. We use these tests to
compare various PRNGs to a QRNG, based on superconducting transmon qutrits and
certified by the Kochen-Specker Theorem, to see whether such evidence can be
found in practice.
  While our tests fail to observe a strong advantage of the quantum random
sequences due to algorithmic properties, the results are nonetheless
informative: some of the test results are ambiguous and require further study,
while others highlight difficulties that can guide the development of future
tests of algorithmic randomness and incomputability.
