In this work we study new ways to observe and characterize specific fractional quantum Hall (FQH) states.

In the first chapter we investigate the possibility to realize specific FQH states in bilayer graphene (BLG). BLG is a novel material in which the electron-electron interaction can be tuned with the help of external parameters. This allows one to make one or another FQH state favourable. We develop a framework for theoretical investigation of the stability of FQH states in BLG. We apply our framework to investigate the stability of the Pfaffian state. We find that the region in which our framework allows for making reliable predictions is quite restricted because of Landau level mixing effects. However, within that region we find the conditions under which the Pfaffian is more stable than in the conventional "non-relativistic" systems. These conditions can, in principle, be realized experimentally.

In the second chapter we focus on characterizing the FQH states with the help of measurements of the noise of the electric current tunnelling between two FQH edges. We develop a theoretical framework allowing for analysing such data, and test it by successfully applying it to describe the results of the experiment [Bid et al., Nature 466, 585 (2010)]. We further develop our framework and show that it is possible to determine the tunnelling quasiparticle scaling dimension from such measurements. We also investigate experimental conditions necessary for this.