I hope this question is not too vague.
For classical bit generators there is the classical statistical definition which (informally) states that a source is ideally random if its output $X_1,X_2,\ldots$ is a sequence of uniform i.i.d. bits.
There is also the alternative Kolmogorov Complexity definition in which an individual string's complexity (informally analogous to randomness) is expressed by its compressibility which is defined as the length of the shortest computer program (w.r.t. a Universal Turing Machine) which will generate the string.
Another definition relates the randomness of the string to the computational complexity of predicting it. This is used, e.g., in the Blum-Blum-Shub generator, where it can be shown that its next output bit can be predicted with probability $(1/2 + \varepsilon)$ for some $\varepsilon>0$ if and only if the RSA problem (given 2 randomly chosen n bit primes $p,q$, form their product $N=pq$ and erase $p$ and $q$; how difficult is it to obtain $p$ and $q$ from $N$?) has an efficient solution polynomial in $n.$
I think some quantum randomness generators use photonic processes to generate bits at their output. However, what is the standard (if any) definition of quantum randomness? I don't know much about quantum theory, but if a quantum system is undergoing the normal Hermitian evolution it seems to me that the output obtained from such a system will not be very random.