Is a quantum system mandatory for generating true random sequence? Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of quantum mechanics involves randomness by definition. Can we argue based on these information that a true random number generator must use quantum mechanics? If anyone claims that s/he has a true random number generator and fails to prove that s/he is exploiting quantum mechanics can I discard the claim on the basis that s/he is not using quantum mechanics?
 A: Classical chaotic systems can be used to generate random numbers.  Specifically, if a system is chaotic then it will have a positive Lyapunov exponent and so will be unpredictable.  Although classical mechanics is deterministic, it is not possible to know the initial conditions to infinite precision.  Therefore it is not possible to predict the future state of a chaotic system.  If the future state cannot be predicted by any means, then it is random.  You would have to make a study of the particular system in order to determine the rate at which randomness is produced, and the best way to extract it, but it can be done.  Chaotic systems are all around us (the weather, turbulence, various electronic circuits, etc.)
However, in a practical sense, it is not possible to do anything without quantum mechanics since you live in a quantum world.  In other words, anything you build in this physical world will, underneath it all, be quantum mechanical.
Now, with quantum mechanics there is something very nice that you can do.  It is in fact possible to build a random number generator in which the numbers produced are certifiably random, even if you do not trust the hardware (say, the hardware was built by your adversary).  For more information on this, search for "Certifiable quantum dice" by Umesh Vazirani (which I have not read).
A: It helps to review Sidney Coleman's Quantum Mechanics In Your Face lecture starting around minute 52:00 and more specifically after min 55:00.  There he talks about the difficulty in determining the randomness of sequences.  Particularly, it is not possible to determine if a finite sequence is random in classical theory, we can only really consider infinite sequences for tests of randomness.
So a test for randomness would look to see if sum value $\sigma$ sums to $0$ at $\infty$.
$$ \lim_{N\to\infty} \bar{\sigma}^N = \lim_{N\to\infty}\dfrac{1}{N}\sum_{r=1}^N \sigma_r = 0$$
as well as seeing if correlations are not present in long strings of data.
$$ \lim_{N\to\infty} \bar{\sigma}^{N,a} = \lim_{N\to\infty}\dfrac{1}{N}\sum_{r=1}^N \sigma_r \sigma_{r+a}= 0$$
for all a
$$ \lim_{N\to\infty} \bar{\sigma}^{N,a,b} = \lim_{N\to\infty}\dfrac{1}{N}\sum_{r=1}^N \sigma_r \sigma_{r+a} \sigma_{r+b}= 0$$
for all a,b
etc.
This situation can be explained in quantum mechanics by asking whether a particular sequence of information can be seen as an eigenstate of an operator with an eigenvalue of zero.
$$ \lim_{N\to\infty} \bar{\sigma_z}^N = \lim_{N\to\infty}\dfrac{1}{N}\sum_{r=1}^N \sigma_z^{(r)} = 0$$
The derivation is:
$$\| \bar{\sigma_z}^N | \psi \rangle \|^2 = \dfrac{1}{N^2} \langle\psi| \sum_{r,s}^N \sigma_z^{(r)}\sigma_z^{(s)}| \psi\rangle$$
$$\langle\psi| \sigma_z^{(r)}\sigma_z^{(s)}| \psi\rangle = \delta^{rs}$$
$$\therefore \lim_{N\to\infty} \| \bar{\sigma_z}^N | \psi \rangle \|^2 =\lim_{N\to\infty} \dfrac{1}{N^2}N=0 $$
which is definitely a deterministic calculation and a random sequence.
IOW, the sequence is random, however the means to derive it is deterministic.  Quantum mechanics itself is a deterministic theory, however the random sequence can be transformed into an observable within the framework of the theory. 
UPDATE
In order to address a criticism that the question was not fully answered.

Is a quantum system necessary if we want to generate true random sequence? 

This somewhat depends on whether one believes there is a quantum explanation behind all natural phenomenon.  What we do know is that a true random number sequence can not be generated with a digital computer.  Computers can only generate pseudo-random numbers which are produced deterministically via some algorithm (with the exception of the possibility that one could conceivably build some type of analog circuit like a mini-lava lamp and then take random numbers from that object and digitize them, unbeknownst to the user).

Can we argue based on these information that a true random number generator must use quantum mechanics?

As explained above, quantum mechanics is deterministic, however random sequences can be understood as observables within the theory.  If you took some amount of a radioactive material and surrounded it with some detection sensors, the particular sequence of emissions detected would be random in nature. 

If anyone claims that s/he has a true random number generator and fails to prove that s/he is exploiting quantum mechanics can I discard the claim on the basis that s/he is not using quantum mechanics?

At a fundamental level the answer is yes.  All processes so far encountered can in principle be described using quantum mechanics (although it is not always convenient to do so).  However, one must be careful in distinguishing the words randomness and unpredictable.  In a classical deterministic system, such as classical mechanics, by definition randomness doesn't exist.  So it is nonsensical in a deterministic system to talk about randomness.  Again, quantum mechanics side steps this particular issue by making states indeterminate before observation.  However, some advocates for super-determinism will argue, even quantum mechanics is fundamentally deterministic and every outcome is still ultimately the result of a "conspiracy". This is arguably an absurd position since it is in some sense tautological.
It is helpful to think in terms of Algorithmic Randomness (Martin-Lof randomness), which can be most simply understood in terms of Kolmogorov complexity where an string of binary digits is considered algorithmically random if it is incompressible.  An equation that can generate a string of pseudorandom numbers is intuitively viewed as a compression of that string, which gets back to the argument about digital computers not being able to generate anything but pseudorandom numbers.
Sidney Coleman's approach, as discussed above, is to show that we can sensible talk about random outcomes and deterministic processes and still be consistent with quantum mechanics. In this sense quantum mechanics is superior to pure classical determinism, which effectively rules out any possibility of true randomness.
A: If you believe in decoherence then the randomness is only apparent and not real. It arises from the interaction with the huge number of degrees of freedom in the environment. So quantum mechanics cannot (by itself) produce a true random number generator.
