Interplay of entanglement structures and stabilizer entropy in spin models

• The manuscript, in its introduction, presents an excellent review of the research field.

• The manuscript investigates a broad variety of one dimensional spin models.

1- Equation 20 and line 259: what are d and dA? Are they the dimension of the Hilbert spaces of total system and of A? These quantities should be defined. 2 - Lines 268-278: this paragraph was hard to follow. Therefore, I would recommend the authors to revise it for improved clarity and flow. 3 - In Section 4: Which bipartition are the authors considering for the computation of entanglement entropy / antiflatness / capacity of entanglement? I would assume it to be half chain, but it should be specified. 4- In the analysis of the XY chain in Section 4.2 the authors report a change in the behaviour of nonstabilizerness and antiflatness moving from the regime of small anisotropy to larger one. I appreciate their argument explaining the transition from localised to non-localised magic. However, I keep wondering what is the physical aspect that makes this shift possible? At first I believed that the oscillations could be related to the oscillations in the parity of the ground-state happening below the separability line, but then one would have to observe similar oscillations also for larger values of $\gamma$. So I wonder if this is, instead, somehow related to the fact that at $\gamma=0.01$ we are very close to the other transition line of the model, and hence we observe the change in complexity?

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1. The introduction provides an excellent pedagogical overview of the research field, offering comprehensive background material.

2. The manuscript presents extensive numerical results on nonstabilizerness across various spin-1/2 models, encompassing several systems that have not been previously investigated in the literature.

1. While this may be inevitable given the nature of this manuscript, the motivation for examining each model and the contributions obtained from such analysis are not clearly articulated, if any.

1. (lines 372 ~) The character $d$ is defined differently from the previous subsection, where it was originally defined as the dimension of the local Hilbert space. Here, it should represent the dimension of the total Hilbert space, i.e., $d=2^n$ for $n$-qubit systems.

2. (lines 511~) The explanation of the phase diagram for the XXZ model is somewhat confusing. In the definition of the Hamiltonian (Eq. (32)), the constant $J$ is introduced. However, the authors give the phase transition lines as $h_s=1+\Delta$ and $h_c$, which should actually depend on $J$. I request revision to clarify the discussion in this section. Additionally, if the spin chain is subject to periodic boundary conditions, this should be explicitly stated.

3. (lines 519~) I do not understand the sentence "In the ferromagnetic phase $\Delta<1$, the SRE is minimal,..." because in Fig. 3(a) the SRE appears to be minimal in the region $0<\Delta$. Perhaps I have misunderstood something, or the horizontal axis in the plots is labeled incorrectly.

4. (lines 549 ~) The authors assume open boundaries for the XY model, but is the classical-like ground state still obtained for open chains? Ref. [134] did not specify the boundary conditions in the model, but later assumed translational symmetry, which appears to be valid only for periodic chains.

5. (page 20 Fig. 8) The legends for the plots are illegible. It would be better to insert the equations manually in the plot area. Another minor comment is that the fitting curves should be moved to the background so that the data points appear in the foreground.

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