Before revision:  One important consequence, by letting k = \log n, is a O(\log^2\log n)-round algorithm for O(\log^{1+o(1)} n) approximation of all pairs shortest path (APSP) in the near-linear regime of local memory.
After revision:  To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an O(\log^2\log n)-round algorithm for O(\log^{1+o(1)} n) approximation of all pairs shortest path (APSP) in the near-linear regime of local memory.