Markle and Steig are presenting quantitative temperature reconstructions obtained from Antarctic ice cores using an improved methodologies which considers the non-linearities in the water isotope/temperature relationships both at the sites (surface and condensation temperatures) and in the precipitation moisture source regions (SST and initial evaporation temperatures). They use a new and relatively “simple” (at least as stated by them) isotopic model, named SWIM, which relies on previously Rayleigh type models (for example the MCIM) and technically explained in a long Appendix, longer than the manuscript itself, although useful for understanding the whole concept. Furthermore, the authors provide a Southern Hemisphere (SH) temperature changes through time, providing Antarctic stacked records, as well as a spatial pattern of LGM-Holocene temperature change in the SH.

The paper calls into questions previously quantitative temperature reconstructions that used isotopic models as well but were not considering the non-linearities embedded in the isotope/temperature relationships.

The paper is interesting, innovative and the topic is appropriate for Climate of the Past.

Nevertheless, the authors should consider some comments reported below before resubmitting a revised version. I will group these comments below and then some detailed and minor/technical comments will follow.

General comments:

- The structure of the manuscript: as reported above, the appendix is very long, and I am not sure that from an editorial point of view this is a good point. However, what explained in the appendix is useful for understanding the SWIM isotopic model and the differences respect to the previous ones. So, I would leave the decision on what to do to the editor. However, I would suggest moving at least 1 or 2 figures from the appendix into the main text, in particular those reporting the differences between previous temperature reconstructions and the one reported in the present study (figure A 27) and the one reporting the main moisture sources for the different ice core sites (A24 or A8).

- The figures: sometimes I found difficult to understand the different colours and, in some cases, for examples for EPICA Dome C the same colour is used for different reconstructions, as in the case of Stenni et al. 2010 and Uemura et al. 2012. Moreover, in some case it is not possible to understand the different ice core records. I believe that this information is needed. Indeed, for EPICA Dome C and probably also for EPICA Dronning Maud Land the old and new reconstructions are quite similar also considering the uncertainties in the reconstruction’s methodologies.

- The impact of these new reconstructions seems to be more important for some sites. Something about this is already mentioned in the text but I would like to visualize better the major differences between previous and this study reconstructions. Perhaps also in the text. See the comments above on the figure colours.

- Regarding the discussion about past elevation changes in West Antarctica from LGM and the Holocene: I would also refer to the Werner et al. (2018 Nature Comm) paper regarding this. Regarding the EDML reconstruction, are the upstream effects considered?

Detailed comments:

Page 10, Figure 5: It would be important to have a legend for the different ice core records. Moreover, some colours are very difficult to see. One record, but I do not know which is (the blue one…) has more positive dxs values at 20 kyr … which is strange…

Page 13, lines 14-19: I do not understand the difference between the absolute and relative uncertainties. Please, may you explain better? An this obviously refers also to Figure 7.

Page 17, lines 9-15: here you are referring to the difference in reconstructions techniques. I would suggest adding here the figure A27 and change the colour for EDC between Stenni et al (2010) and Uemura et al. (2012). See also my comments above.

Page 17, lines 17-27: in this paragraph you are referring to other T reconstruction techniques. What about a recent paper by Buizert et al. 2021, of which one of the authors of the present study, is co-author? Also considering elevation changes effects and reporting quite different cooling during LGM than the ones reported here as well as previously by also other authors. Perhaps a comment on this would be the case, or here or also later in the discussion paragraph, also referring to Figure 11.

Page 42, lines 6-7: this is in contract with observations in precipitation at Concordia station (Stenni et al., 2016) where mean annual precipitation weighted isotopic values are less negative than arithmetical means and temperatures are warmer ….. if I understood correctly.

Page 42, lines 28-29: in Masson-Delmotte et al. most of the samples are from surface snow (or mean of firn shallow cores) rather than precipitation.

Minor and technical comments:

Page 1, line 16: Change “Barbante et al.” in “EPICA Community Members”; change also in the References.

Page 1, line 19: add “*10³”;

Page 14, line 7: add “of” between “function” and “reconstructed”.

Page 15, figure 8: the grey lines are not visible, and the same for light grey and thin dark grey ….

Page 16: figure 9: it is not possible to see in a clear way, moreover no way to understand to which ice core records you are referring.

Page 18, lines 8-10: I would suggest to add here something more about elevation changes ….. see paper from Werner et al 2018.

Page 19, figure 11: also here I had some difficulties with the colours in panel a).

Page 21, lines 1….. again the comment above on elevation changes

Page 21, lines 11-12: the linear definition of dxs is an unreliable …. At Dome C this doesn’t seem the case..

Page 26, line 22: I suppose that “complication” is compilation…

Page 32, line 10: I suppose that “modification” is fractionation.

Page 58, line 21:I suppose that one of “Ts” is Tc.

Page 61, line 1: please, change “EPICA Dome Concordia” into EPICA Dome C.

Page 63, figure A27: see my comments above regarding EDC (please use different colours for Stenni et al and Uemura et al). Please check also EDML for upstream corrections.

Page 63, lines7-9 and also page 64 lines 1-2: there are two sentences that are repeated.

General comments:

This manuscript describes the improved method for reconstructing Antarctic temperature based on ice core water isotope record. Although the manuscript is very long (70 pages), it is organized well and easy to read. The previous temperature reconstruction methods based on the Rayleigh-type model has been properly improved and many potential uncertainties and/or assumptions (closure assumption, inversion temperature, mixing of air mass etc.) are carefully evaluated. Overall, this manuscript is very interesting and suitable for publication in Climate of the Past. To improve the manuscript, I made some comments below.

Major comments

(1) Abstract "..However, there are important nonlinearities that significantly affect such reconstrugion..Here, we describe a temperature reconstruction method that account for these nonlinearities.."

I think the Abstract (and main text) overemphasizes only the difference between linear and non-linerar reconstructions. The difference between this and previous studies results from not only linear/non-linear technique but also different settings (evaporation, supersaturation, etc.) used for isotope modelling. I think the important contribution of this study is the careful examination of various factors one by one, which surely improve the understanding of uncertainty of several important assumptions. I think it is better to write this point in the abstract. In fact, the authors themselves noted that "The difference between the results of this study and the previous temperature reconstructions arise from differences between the linear and nonlinear reconstruction technique as well as differences in the underlying water-isotope models used for the estimation of scaling relationship".

(2) Evaporation from the ocean (Appendix A2.1).

P27 L15 ".. this "local" closure assumption.. "

> Since Merlivat and Jouzel (1979) assumed a global steady state of water cycle, the assumption (Rv=Re) has been commonly referred to as "global closure assumption". But, here, the authors termed this as "local" closure assumption. To avoid unnecessarily confusions, it is better to add short explanations about different terminology.

By the way, this section (A2.1) is interesting and includes important analyses. In fact, the different assumptions affect the T reconstructions (Fig A21). So, I think this section, at least some part, may be moved to main text.

(3) P20 L1-6 "We find smaller glacial-interglacial temperature change for East Antarctic sites compared to previous reconstructions .. The average warming at the two highest sites however, Dome Fuji and Vostok, is significantly less, just 6.9 degC or 59% of that at the lower sites."

> Previous T reconstructions at DF and Vostok are 7.5-7.8 deg C (e.g., Vimeux et al. EPSL 2002, Uemura et al., CP 2012) (depending on time intervals you choose). Thus, the difference between previous and reconstructions is only 0.6-0.9 degC, which is not significant. For objective comparison, it is better to describe the exact differences from past reconstructions and add short descriptions.

(4) P53. L11-13 "such as the value of the diffusive fractionation factors during transport.."

Does your model include the eddy diffusive fractionation during transport, like described in Hendrick et al. (GBC, 2000)? If so, please describe it in the Appendix.

Hendricks et al., GBC, https://doi.org/10.1029/1999GB001198

(5) Difference between delta-age based temperature.

A recent paper by Buizert et al. (2021) claimed that Antarctic temperature during LGM is less than those estimated with the water isotopes. And they suggest that the difference can be attributable to an altered Antarctic temperature inversion during the LGM. This finding is very closely related to the topic of this manuscript. Maybe this is beyond the scope of this paper. I would like to ask about some comments about the differences between this and Buizert et al. (2021) (actually, the second author of this manuscript is a coauthor of Buizert et al. 2021).

Buizert et al. (2021) Antarctic surface temperature and elevation during the Last Glacial Maximum, Science, 10.1126/science.abd2897

Specific comments

Figure 5 left panels > I don not see each profile corresponds to which ice core. Please add appropriate legends in left panels.

Figure 8 Legends > Please sort in order of condensation temperature, to make it easier to compare with color profiles in panel c and d.

Figure 9 panel a and b > I don not see each profile corresponds to which ice core.

P8 L5-6 ".. and the changes in slopes across the parameter space are larger than these local changes.."

>I do not understand this sentence.

P17 L3. " .. colder temperature then previous studies.."

> than

P21 L21-28 "There is a long-standing debate regarding the interpretation of "spatial" and "temporal .."

> Since this topic is not discussed in the main text, it seems strange to suddenly come up with a conclusion. How about moving this topic to the introduction?

P36 L31. ".. that condensation occurs at aa range of temperature up to.."

> a

Figure A23 > Colored profiles cannot be recognized.