We thank Reviewer (Nun8) for their response. We are happy to hear that they agree the $n\geq 1$ setting is interesting and that the extension is non-trivial.
We also find that there seems to be no known lower bound for blackboard or sequential protocols when $n > 1$ in the case of testing. In [2], the authors show that for $n=1$, there is no benefit to a sequential setup when compared to a shared randomness setup for uniformity testing with discrete data. For the Gaussian location model, we note that the proof of the shared randomness lower bounds of [11] can be extended to obtain the same (rate) results for sequential setups as well. Our (extended) machinery of the revised version then explicitly implies that there is no benefit of a sequential setups (outside of shared randomness) for uniformity testing with discrete data in the regime where $md \log (d) / \sqrt{n} = o(1)$. We will add details of this particular extension to the revised version of our paper.
Admittedly, this does not mean that we can conclude anything concerning the benefit of a sequential setup in the regime(s) where $n > 1$ and $md \log (d) / \sqrt{n} \gtrsim 1$. Also, we note that for blackboard protocols, much less is known and we can indeed not exclude to benefit of a blackboard protocol. Proving lower bounds in the testing setting for blackboard protocols specifically is an interesting but difficult problem with many open questions in the literature.
We will also include a discussion on why the technique of [1] does not yield an optimal lower bound in the testing setting in our revised version. We agree that including such a discussion is important to highlight the contribution of our work. We briefly sketch the reasons that this (and certain other) estimation techniques do not extend to the testing setting below.
Let us start with describing a similarity: for both the estimation and the testing problem, lower bounds are typically proven by bounding a divergence measure between probability distributions, such as the chi-square divergence, mutual information or total variation [1,3,7,11,12,13,18,19].
For estimation problems such as the one considered in [1], or those of the examples considered in [3], it suffices to essentially "tensorize" the divergence, which, loosely speaking, breaks the problem into the ``sum of the local divergences''. This is essentially the role of Theorem C.2 in [1] (see also Theorem 1 and 2 in [3]), which bounds the total variation between elements of a perturbed family of probability distributions by the sum of the local conditional "scores", local conditional variances of the transcript densities or the local mutual information, see (16), (17) and (18) in [1]'s supplement. The loss due to a bandwidth constraints (in [1]) or privacy constraint (in [3]) is then captured by data processing arguments. For estimation, such tensorization bounds turn out to give tight lower bounds. We note that a lot more goes into the proof of [1] (e.g. Poissonisation, sub-Gaussian concentration), but the principle difference with estimation and testing is this kind of tensorization step. Similar tensorization arguments can also be found in other estimation problems such as [4,7], for the Fisher information and mutual information respectively.
Such a "tensorization approach" does not yield tight bounds in testing problems. Using mutual information, [19] tries this tensorization approach for the testing problem, but they only recover the optimal testing rates when each server communicates only one bit ($b=1$). Another "estimation" approach tried for a goodness-of-fit testing problem can be found [18], which similarly obtain a lower bound for testing that is only tight for $b=1$, through a direct Taylor expansion of the likelihood (which can also be seen as tensorizing a divergence). The authors of [18] provide a detailed discussion of the shortcomings of the latter approach in Section 4 of their paper.
To obtain tight lower bounds for the testing problem, the papers that successfully do so for $b > 1$ and privacy constraints (i.e. [11,12,13]) use techniques that differ greatly from the techniques employed in [1,3]. In [13], the authors use a combinatorial expansion of the likelihood that works specifically for $n=1$ in the multinomial model, which does not generalize to large numbers of observations. [11,12] circumvent the latter issue in the Gaussian setting by employing a Brascamp-Lieb inequality, an inequality from functional analysis. This inequality explicitly uses the Gaussianity of the log-likelihood explicitly.
We hope that above additions to our work are satisfactory and we would like to thank you again for your consideration of our work.
Additional references:
[18] Acharya et. al., "Distributed signal detection under communication constraints"
[19] Szabo et. al., "Optimal distributed composite testing in high-dimensional Gaussian models with 1-bit communication"