Cirel'son investigates quantum generalizations of Bell's inequality, showing that quantum correlations, while violating Bell's inequality, satisfy weaker inequalities. He proves specific inequalities and discusses the general case of quantum measurements in different space-time regions. The paper presents four theorems characterizing quantum correlations between observables and parameters. The first theorem states that for observables with spectra in [-1, +1], certain conditions are equivalent. These conditions involve quantum algebras, density matrices, and specific operator properties. The theorem also shows that for m = n = 2, operators can be chosen as 2x2 matrices with specific properties, allowing for maximum spin correlations. It is shown that the maximum value of a particular combination of spin correlations is 2√2, which is greater than the classical limit of 2. This implies that the inequality c₁₁ + c₁₂ + c₂₁ - c₂₂ ≤ 2√2 holds for arbitrary quantum observables. The paper also discusses the physical implications of these results, showing that quantum mechanics allows for stronger correlations than classical theories. The theorems are proven in detail, with the proofs published elsewhere. The paper concludes with a discussion of the implications of these results for quantum mechanics and the nature of quantum correlations.Cirel'son investigates quantum generalizations of Bell's inequality, showing that quantum correlations, while violating Bell's inequality, satisfy weaker inequalities. He proves specific inequalities and discusses the general case of quantum measurements in different space-time regions. The paper presents four theorems characterizing quantum correlations between observables and parameters. The first theorem states that for observables with spectra in [-1, +1], certain conditions are equivalent. These conditions involve quantum algebras, density matrices, and specific operator properties. The theorem also shows that for m = n = 2, operators can be chosen as 2x2 matrices with specific properties, allowing for maximum spin correlations. It is shown that the maximum value of a particular combination of spin correlations is 2√2, which is greater than the classical limit of 2. This implies that the inequality c₁₁ + c₁₂ + c₂₁ - c₂₂ ≤ 2√2 holds for arbitrary quantum observables. The paper also discusses the physical implications of these results, showing that quantum mechanics allows for stronger correlations than classical theories. The theorems are proven in detail, with the proofs published elsewhere. The paper concludes with a discussion of the implications of these results for quantum mechanics and the nature of quantum correlations.