LS minimizes mean squared error to solve the SVM regression problem. In SVM classification, coefficients alpha are bounded between 0 and C (the box constraint). Support vectors (observations with positive alphas) obtained by solving the classification SVM problem are a fraction of the training set. For minimization of MSE, coefficients alpha are not bounded, and most usually every observation in the training set gets a non-zero alpha coefficient.
QP solves the classification problem but is slow on large datasets. For large-scale classification problems, your best bet is SMO.
