Just out of curiosity: Given a "black box" implementation of a random number generator, is it possible to test its output sufficiently to gain some faith in its proper randomness?
gnasher719 gave a nice answer, and I just want to add more. For statistical purposes, a true and fair RNG printing a string of zeroes and ones would at the very least print a normal number, and to test for that, one would have to count the frequencies of substrings of every length, in every base, which of course can be done, but may get expensive if one wants a lot of confidence.
What gnasher719 calls cryptographical randomness cannot be tested in practice, but in theory, one can run the countable set of all computer programs in parallel, and see whether they can predict the digits of the RNG sequence with probability better than 1/2. Of course, this is completely intractable in the foreseeable future, with or without functional quantum computers.