We consider the “space” of supersymmetric conformal field theories (SCFTs) in 4d. Many,  conjecturally all, 4d SCFTs can be engineered by  reductions of 6d SCFTs on Riemann surfaces. Thus the space of 4d SCFTs can be covered by overlapping patches labeled by 6d SCFTs. We will first review  how one can associate a system of commuting relativistic  quantum mechanical (QM) Hamiltonians  to such patches and the relation of these systems to supersymmetric index computations. Next, we will discuss examples of non-relativistic limits of the QM models and the corresponding indices. We will claim that the further free limits of these non-relativistic models can be thought of as generalizations of the powerful N=2 Schur index to the more general N=1 setups.  We will discuss some of the surprising properties these “free” indices satisfy: e.g. in some cases the indices of theories down the RG flow “know” information about the theories up the RG flow which naively they do not entitled  to know. The talk is based mainly on https://arxiv.org/abs/2604.19885 and https://arxiv.org/abs/2506.13764.