Dear Editor,

we again like to thank the Referee for the careful analysis of the manuscript. To address the issues in order:

C1) We certainly believe that our figure 1 is a helpful illustration of the comparison between an actual measurement, and a Universal Turing Machine thatbis at the core of the argument of this work, and feel it is, with all its imperfections in a completeness of a mapping between the two scenarios, a helpful guide for a reader on what problem we try to address. Therefore, we really want to keep it, especially for those who have a more graphical approach in reading a paper - even if one can imagine cases where such a representation would be problematic.
So we appreciate the concerns the referee has with the incompleteness of a graphical representation, but feel this does not justify dropping it. As for the the creative model proposed by the Referee, it does not does not violate inequality (6), as indicated. It also does not violate the entropic inequality (1). As such, we do not find relevant that it does not comply with UC, even if there are, and again we agree with the referee there, most likely more creative models that violate UC.

It would be very interesting to find a model violating inequality (6) that does not conform to UC, and it would be especially interesting to compare it with the case of a standard Bell test. As of now, we have not been able to find any such model for non-FSSO experimental tests. In fact, we believe that UC is an assumption necessary *only* in the case of FSSO setups, but we have not been able to find a proof strong enough to support this claim. For this purpose, the UC assumption is introduced at the end of section 2.2, as a necessary condition for the experimental repeatability, a necessary condition for every scientific theory. This does not contradict any part of the manuscript. We are confident that the demonstration that the violation of (6) implies that "either local realism or uniform complexity are invalid" is as robust as the demonstration for the standard Bell inequality for the entire class of i.i.d. local hidden variable models.


Figure 1, within the limits of a graphical representation, therefore represents a model for possible local variables theories, including models not conforming with UC, as described in section 2. The generic bit strings indicate the type of input and output obtained from a UTM do not represent the corresponding Kolmogorov complexity.

As a matter of fact, also the model proposed by the Referee is represented by figure 1:
- UTM A and B are provided a program and a string containing measurement.
- Each UTM produces a resulting string with length equal to the input measurement string. 
Since the Referee explicitly cites the memory of the device as distinguishing factor, we would like to point out that a UTM, in their abstract definition, does not have any memory constraint, i.e. all possible memory-based schemes are included.

To address the Referee's reservations about the impact of UC on the generic model, we have modified the last paragraph of section 2.2 to make it explicit the relevance of UC, and its relationship with the usual i.i.d assumption in a CHSH test. We think we have also clarified the different meanings of indexes i and i', j, and k better. Additionally, we added a statement that we assume uniform complexity at the end of the second paragraph of the conclusion section.
 

C2). The last paragraph of section 3.1 has been rewritten to clarify the relationship between entropic inequality and inequality (6) for the case of a bipartite maximally entangled state. 
We have also clarified that the maximal violation calculated is relative to the geometry introduced by Braunstein and Caves.

We have restricted our tests to this model because it gave already a large enough range of experimentally  accessible parameters where we do see a violation, and simplified the experimental data taking to to an easily accessible one-dimensional search. Certainly we could probe a larger parameter space, but we feel it would not add much to the main idea of the paper.


Follow-up on typos and grammatical errors:
- Page 2, section 1, third paragraph: from "[...] conditional the [...]" changed to "[...] the conditional [...]", "H(ab)" changed to "H(a|b)"
- Page 2, Eq. (1): corrected the subscripts
- Page 3, section 2, first paragraph: from "The algorithmic approach we present, instead, considers[...]" to "The algorithmic approach we present instead considers[...]"
- Page 3, section 2, second paragraph: removed "to"
- Page 6, near end of paragraph after Eq. (7): corrected the reference to equation (1)
- index range for i,k: the clean-up of equation 1 and section 2 should have fixed this problem.
- fixed a spelling error "withing" -> "within" in section 5.1
- updated reference [24] from arXiv to published paper

With this, we hope to have addressed the remaining concerns of the referee with this manuscript.

With Best Regards on behalf of all authors,

Christian Kurtsiefer