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1. It is not a CHSH test. As pointed out in the text, the inequality studied is directly connected to the entropic inequalities introduced by Braunstein and Caves~\cite{Braunstein:1988en}.
As indicated in the introduction, the novelty is in the use of a metric based on the complexity of a string instead of the statistics of the outcomes to study the limits of local realism.

2. The referee probably is not familiar with the difference between non-compressible and non-computable. He also missed how the non-classicality of the system is evident only when considering two separate measurement, as explained in section 1.2 of the manuscript.
In the manuscript we clearly explain how we study a bi-partite system, beyond the definition the simple idea suggested by the referee.

3. This referee rises an interesting point. So interesting that we have included in the manuscript, not only a reference to seminar works on NCD and its connection to Kolmogorov complexity (Cilibrasi R and Vitanyi P M B, 2005 Information Theory, IEEE Transactions on 51 1523–1545), but we have also included section 3, entirely dedicated to the study of how different compression algorithms affect the approximation of NCD to Kolmogorov complexity.

4. The referee's comment seems to imply that inequality (4) is violated while inequality (3) is not. We would like to stress how there is not any assertion of this kind in the manuscript. We state how the triangular inequality is used to transform inequality (3) into inequality (4).
Similar inequality relations are commonly used in the scientific literature, and are fundamental of algebra. We are thus surprised by the comment of the referee and we are not sure that a practical example could help much in understanding this basic relations.
We would also like to emphasize that the need for the derivation of ineq. (4) is due to the physical origin of the bit strings, as indicated in the manuscript: "$\NID(y_{b_0}, y_{b_1})$ cannot be determined experimentally because the strings $y_{b_0}$ and $ y_{b_1}$ come from measurements of incompatible observables."

5. We have tried to clarify the text, addressing some of the points raised by the referee, in the hope that he could eventually understand what is the main point of the manuscript:
- The manuscript does not set bounds on the compressibility of measurement outcomes. as stated in the abstract: "We experimentally demonstrate an impossibility to reproduce quantum bipartite correlations with a deterministic universal Turing machine."

- From the manuscript: "We now extend this picture to two spatially separated UTM’s UA (Alice) and UB (Bob). If these machines cannot communicate, they generate two output strings that are independent, although the programs fed into the machines can be correlated."
As indicated in this sentence, the machines are fed different programs. These programs can be correlated. The main point of the manuscript is that the correlation that can be observed at the output of the two UTM is classically bounded by the correlation between the inputs, i.e., the NID of the output strings.
- We would like to remind the referee that a program fed into a machine can also contain indication about which machine should execute a specific instruction. it is thus possible to feed the same program to two independent UTM and obtain two different strings. This is exactly the scenario that we are presenting.
- By definition, every string can be reproduced by a UTM and a suitable program. This definition does not require further assumptions, apart from the existence of UTM.
- We rephrased the statement: ---
- We thank the referee for the suggestion. We notice that the reference indicated does not make any reference to the complexity of the output generated by quantum measurement. It instead shows that computers impose a limitation when it comes to producing a mixed state as a classical mixture of pure quantum states, and the implication of how a classical Eve can mimic a Bell inequality violation when the measurement choices on Alice and Bob are performed following an algorithm.

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