Question

Given a stream of random binary numbers(*)

Is there any way to differentiate if they came from a Truly Random or from a formula/algorithm ? how?

if there is no way to decide this, then, I can't find any basis, to keep denying that behind the "truly random" of quantum mechanics can be a hidden algorithm.

I know I am talking about the posibility of a "hidden variables" theory, but I can't find any other explanation.

(*) Is known that is possible to create computable normal numbers from which first is possible to extract an infinite random binary stream, second, it tell us that a finite logic expression (again a binary stream) can contain a rational, and even an irrational number, so there is no big difference within a bit stream and the measurement of a random outcome, (and even less if we consider the accuracy) I say this, because the argument that obtaining bits is not the same as answer the complex question that can be made to quantum experiments, I think that "random" results can be perfectly read from a random stream of bits

Answer 1

This question involving "randomness" and Quantum Mechanics introduces some subtleties. Firstly we have the definition of "randomness" to consider. It turns out that there are various ways to define this term: the two I shall consider here are (the digit sequence of a) normal number (mentioned in the Question) and Martin-Lof Randomness.

As explained in the Wikipedia article a "Normal Number" is "Finite State Machine Random". So it looks random to a FSM. However such numbers can be computable by a Turing Machine (as the link shows that Turing proved).

A Martin-Lof random sequence is based on the familiar notion of incompressibility and cannot be computed by a Turing Machine. Phrased alternatively because it is an infinite sequence it will not have a finite compression onto a Turing Machine - so there can be no program for it (which has to be finite).

The logical link between the two is that every Martin-Lof random sequence is normal (but not conversely - as shown by Turing).

Also note that every finite sequence can be generated by an algorithm (and also it will be the solution of a polynomial). The reason why the random sequences work is that although the first N digits can be replicated via a program P, a different program (in general) is required for the first N+1 digits of the sequence ie P will fail to "predict" N+1. To get the entire sequence requires an infinite series of programs so we are back where we started with the random sequence.

Now that we have some definitions available we can examine connection to Physics. The problem in a experimental based theory is that we only ever have a finite amount of data. Thus the claim that a given series is "random" is strictly not empirically provable.

So this puts the Copenhagen claim that "QM sequences are random" into an awkward semi-scientific status. Such claims cannot be proved experimentally, yet Copenhagen asserts this. So where is the proof? Indeed what constitutes a proof? Also we have seen several definitions of randomness (there are more) - so which type of randomness does QM have exactly?

To return to specific points in the OP question:

Is there any way to differentiate if they came from a Truly Random or from a formula/algorithm ? how?

No, because one only ever has a finite amount of stream data to analyse, which can always be explained algorithmically (as discussed above).

if there is no way to decide this, then, I can't find any basis, to keep denying that behind the "truly random" of quantum mechanics can be a hidden algorithm.

The "truly random" of quantum mechanics might be Martin-Lof randomness, for which there is no algorithm; however if it is really normal number randomness then there might be an algorithm. I suspect that most physicists take the view that QM is as "random as it gets", hence would prefer the Martin-Lof option.

I know I am talking about the possibility of a "hidden variables" theory, but I can't find any other explanation.

The link between algorithmic underlying structure and "hidden variables" would appear to be close. If its a non-algorithmic type of randomness then we have to decide whether it belongs to one of "oracle classes" associated with Martin-Lof randomness, and what that would mean in terms of "hidden variables".

Some readers might recall that in his book "The Emperor's New Mind" in 1989 Roger Penrose proposed that aspects of quantum mechanics were "non-computable". Although that argument was not formulated as I have here, it is consistent (I believe) with the idea of Martin-Lof randomness too