Estimating the spectrum of a matrix is a fundamental problem in numerical linear algebra. In this talk, we explore the efficiency of estimating the spectrum of a symmetric matrix with bounded entries using entrywise sampling. Prior work established that an eps * n additive approximation to all eigenvalues could be achieved by sampling a principal submatrix of size poly(log n) / eps^3. We show that a sample size of O~(1/eps^2) is sufficient, removing the dependence on n and obtaining optimal dependence on eps up to log(1/eps) factors. We extend these techniques to achieve tight bounds for related problems. We show that an eps * ||A||_F additive approximation via squared row-norm sampling requires only O~(1/eps^2) samples, improving the previous O~(1/eps^8) bound. Furthermore, for bounded entry PSD matrices, we prove that sampling O(1/eps) columns allows for a non-adaptive approximation of the top eigenvector to within eps * n additive error. Finally, we discuss applications of these results, including faster eigenvalue estimation sketches for dense matrices and improved sample complexity for approximating the spectrum of bounded entry PSD matrices. Based on joint work with William Swartworth