In this paper, we propose several new constraint qualifications for mathematical programs with second-order cone complementarity constraints (SOCMPCC), named SOCMPCC-K-, strongly (S-), and Mordukhovich (M-) relaxed constant positive linear dependence condition (K-/S-/M-RCPLD). We show that K-/S-/M-RCPLD can ensure that a local minimizer of SOCMPCC is a K-/S-/M-stationary point, respectively. We further give some other constant rank-type constraint qualifications for SOCMPCC. These new constraint qualifications are strictly weaker than SOCMPCC linear independent constraint qualification and nondegenerate condition. Finally, we demonstrate the relationships among the existing SOCMPCC constraint qualifications. [ABSTRACT FROM AUTHOR]