It has been a longstanding conjecture in entanglement theory that all PPT states in 3x3 dimensions have Schmidt number less than 2. Recently, this conjecture has been proven by Yang et al. Motivated by this result we study Schmidt numbers of bipartite PPT states in higher dimensions. We start by presenting an explicit construction of PPT states achieving Schmidt numbers scaling linearly in the local dimension improving on previous constructions. Next, we link the Schmidt number of a quantum state written as a block matrix to entangled sub-block matrices. We use this to show that states invariant under partial transposition on the smaller of their subsystems cannot have maximal Schmidt number. This generalizes a well-known result by Kraus et al. Finally, if time permits we discuss applications of these techniques to problems of channel compositions and most importantly to the PPT squared conjecture. The first part of the talk is joint with M. Huber, L. Lami and C. Lancien, and the second part with M. Christandl and M. Wolf.