This paper studies the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. The authors establish near-optimal SQ lower bounds for NGCA under the moment-matching condition alone, without requiring the chi-squared distance restriction. Specifically, they prove that any SQ algorithm solving the NGCA problem with a distribution \( A \) that approximately matches moments with the standard Gaussian up to degree \( d \) must either use a query with accuracy \( O_{m,d}(n^{-(1-\lambda)/4-c}d) \) or at least \( 2^{n^{\Omega(c)}} \) many queries. This result generalizes to the setting of a hidden subspace and has implications for various learning problems, including list-decodable Gaussian mean estimation, anti-concentration detection, and learning periodic functions. The proof techniques involve Fourier analysis and Hermite polynomials, addressing technical challenges such as the unbounded chi-squared norm and the interchange of summations.This paper studies the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. The authors establish near-optimal SQ lower bounds for NGCA under the moment-matching condition alone, without requiring the chi-squared distance restriction. Specifically, they prove that any SQ algorithm solving the NGCA problem with a distribution \( A \) that approximately matches moments with the standard Gaussian up to degree \( d \) must either use a query with accuracy \( O_{m,d}(n^{-(1-\lambda)/4-c}d) \) or at least \( 2^{n^{\Omega(c)}} \) many queries. This result generalizes to the setting of a hidden subspace and has implications for various learning problems, including list-decodable Gaussian mean estimation, anti-concentration detection, and learning periodic functions. The proof techniques involve Fourier analysis and Hermite polynomials, addressing technical challenges such as the unbounded chi-squared norm and the interchange of summations.