Because neither referee addressed the scientific content of our paper,
nor substantively addressed its presentation, we have not submitted a
revised manuscript. Below, we respond to the referees' comments.

Report of Referee A -- LY17256/Kent-Dobias
> The authors consider spin glass models with mixed p-spin interactions
> on the N-Sphere and calculate the number of stationary points, the
> logarithm of which yields the complexity. The disorder average of this
> logarithm is computed with the replica trick, and for different model
> variants different replica symmetry breaking (RSB) solutions are
> obtained. A new feature of the solutions, in contrast to previous
> replica symmetric calculations, is that RSB must occur in parts of the
> energy-stability phase diagram.
>
> The paper is clearly written although the content is rather technical
> and probably only accessible to experts in mean field spin glass
> models and the different RSB schemes developed in this field. In
> connection with the well-studied p=3 spin glass model it is briefly
> mentioned that the complexity and its transitions as a function of
> energy and/or stability is relevant for the equilibrium and the
> dynamical behavior of this model – but such a connection has not been
> made here.
>
> Therefore, I feel that the results presented here are only interesting
> for group of experts and I do not assess the finding that the
> complexity of mixed p-spin glass models shows RSB as a major
> breakthrough in the field. Therefore, the manuscript is not suitable
> for publication in Phys. Rev. Lett., and the publication of the
> accompanying longer paper, submitted to PRE, is sufficient to
> disseminate the results summarized in this manuscript.

Referee A says that our paper is clearly written but technical, and
that its topic of "the different RSB schemes" are not suitable for a
broad audience. This is surprising to the authors, since a quick
search on Google Scholar reveals several recent PRLs with heavy use of
RSB schemes.

Referee A correctly points out that one new feature of the solutions
outlined in our manuscript is that RSB must occur in parts of the
phase diagram for these models. However, they neglect another feature:
that they are the solutions of the *quenched* complexity, which has
not been correctly calculated until now. We agree with the referee
that "the complexity of the mixed p-spin glass models" is not a major
breakthrough in and of itself, but believe that the technique for
computing the quenched complexity is a major breakthrough. Just
because this new technique is demonstrated on the simplest toy models
should not discount its importance and potential.

Referee A states that a connection between the complexity and the
equilibrium and dynamical behavior is not made in our paper. Until
recently, this connection was taken for granted, and the demonstration
that the standard correspondence does not hold in the mixed p-spin
spherical models was exciting enough news to be published in PRX 10,
031045 (2020). It is true that our work doesn't solve the problem that
paper opened, but it does deepen it by showing definitively that the
use of RSB and the quenched complexity are not sufficient to
reestablish a landscape–dynamics connection. We disagree with the
referee's implicit assertion that only clean resolutions, and not the
compelling deepening of problems, are worthy of a broad audience.

Report of Referee B -- LY17256/Kent-Dobias
> The paper presents a computation of the complexity in spherical
> spin-glass models. Neither the techniques nor the results are
> sufficiently new and relevant to justify publication on PRL. This is
> not surprising given that the topic has been studied extensively in
> the last thirty years and more, the only novelty with respect to
> previous work is that the results are obtained at zero temperature but
> this is definitively not enough. Essential open problems in the field
> involves dynamics and activated processes and some results have
> appeared recently, instead the analysis of the static landscape, to
> which the present paper is a variation, failed to deliver answers to
> these questions up to now.
>
> I recommend that the paper is not published in PRL while the companion
> paper is worth publishing on PRE.

We disagree with the statement of Referee B that "the only novelty
with respect to previous work is that the results are obtained at zero
temperature". For a system where the quenched and annealed
complexities differ, there has not been a correct calculation of the
quenched complexity at finite temperature. (and, besides our work,
only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).)
Rejecting a paper based on a severe misconception of its contents or
of the state of the field is not appropriate.

We agree with Referee B's assessment of "[e]ssential open problems in
the field," and agree that our work does not deliver answers. However,
delivering answers for essential open problems is not the acceptance
criterion of PRL. These are

  - Open a new research area, or a new avenue within an established area.
  - Solve, or make essential steps towards solving, a critical problem.
  - Introduce techniques or methods with significant impact.
  - Be of unusual intrinsic interest to PRL's broad audience.

We believe our manuscript makes essential steps toward solving the
critical problem of connecting analysis of the static landscape to
dynamics. We believe that its essential step is through the
introduction of a new technique, calculation of the quenched
complexity, which we believe will have significant impact as it is
applied to more complicated models.