Correlated phases in Hall systems have topological character. Multilayer configurations of planar electron systems create the opportunity to change topological phases on demand using macroscopic factors, such as vertical voltage. We present an analysis of such phenomena in close relation to recent experiments with multilayer Hall setups including GaAs and graphene multi-layers. The consequences of the blocking or not of the inter-layer electron tunneling in stacked Hall configurations are analyzed and presented in detail. Multilayer Hall systems are thus tunable topological composite nanomaterials, in the case of graphene-stacked systems by both intra- and inter-layer voltage.
Keywords: FQHE hierarchy in Hall systems; bilayer Hall systems; control over homotopy phases; correlated Hall homotopy phases; indirect excitons; long-range quantum entanglement; reentrant IQHE; superfluidity of excitons.