We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set of gradients of inequality constraints whose properties have to be preserved locally. MPEC-RNNAMCQ and MPEC-CRSC have nice properties: they have the local preservation property and imply the error bound property under mild assumptions. Thus, they can be used to extend some known results on perturbation analysis and sensitivity. Both conditions can also be used in the convergence analysis of several methods for solving MPECs. Relations to other MPEC-CQs will also be discussed.