Sequential optimality conditions have recently played an important role on the analysis of the global convergence of optimization algorithms towards first-order stationary points and justifying their stopping criteria. In this paper we introduce the first sequential optimality condition that takes into account second-order information. We also present a companion constraint qualification that is less stringent than previous ones associated to the convergence of second-order algorithms, like the joint condition Mangasarian-Fromovitz and Weak Constant Rank. Our condition is also weaker than the classical Constant Rank Constraint Qualification, which associates this condition to the convergence of second-order algorithms. This means that we can prove second-order global convergence of well stablished algorithms even when the set of Lagrange multipliers is unbounded, which overcomes a limitation of previous results based on MFCQ. We prove global convergence of well known variations of the augmented Lagrangian and Regularized SQP methods to second-order stationary points under this new weak constraint qualification.