-Comparing the complexity analysis results of these protocols, we can see that the unoptimized zeromorph protocol has the largest msm operation length, $n ~ \mathsf{msm}(D_{max} + 1,\mathbb{G}_1)$, and the largest proof size, $(2n + 1) \mathbb{G}_1$. The detailed complexity analysis can be found in [zeromorph-anlysis](https://github.com/sec-bit/mle-pcs/blob/main/analysis/zeromorph-anlysis.zh.md). Both zeromorph-v1 and zeromorph-v2 adopt different methods to optimize the degree bound proof, reducing the Prover's msm operations and decreasing the proof size by about $n ~ \mathbb{G}_1$. The main difference between zeromorph-v1 and zeromorph-v2 is that zeromorph-v2 avoids Verifier operations on the elliptic curve $\mathbb{G}_2$, at the cost of increasing constant-level computation for the Verifier on the elliptic curve $\mathbb{G}_1$ and a proof size of $\mathbb{G}_1 + \mathbb{F}_q$.
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