Is there a standard definition of Quantum Randomness? 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 $\varepsilon&gt;0$ 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.
The randomness used in quantum computing comes from quantum mechanics and is postulated to be an inherent 'randomness' of nature (and one of the problems Einstein, for instance, had with QM). If you take a state like

$|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$ 

and measure it in the computational basis, there is absolutely nothing you can do to predict the result (assuming our standard pictures of QM is not horribly wrong). Half the time you will measure $0$ and half the time you will measure $1$. Of course, if you measure in the Hadamard ($|+\rangle$, $|-\rangle$) basis, then your result will always be $+$.

I don't think there is a standard notion of quantum randomness, mostly because how random a state looks depends on your choice of measurement. Once you choose a particular basis to measure in, then you might as well start talking about the classical probabilities associated with the results and then you can do your standard classical tests of randomness.

Usually when someone talks about the 'quantumness' of a state, they try to capture ideas like entanglement.