The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this talk, I consider a simple bilayer fractional quantum Hall (FQH) system with the 1/3 Laughlin state in each layer, in the presence of interlayer tunneling. I show that interlayer tunneling can drive a continuous phase transition to an exotic non-Abelian state that contains the famous `Fibonacci' anyon, whose non-Abelian statistics is powerful enough for universal TQC. The analysis that I will present towards this result rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions.
Next, I discuss the experimental aspects of our proposal and potential probes for the Fibonacci phase. I show that the charge gap remains open at the phase transition while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at nu=2/3 in bilayers, as past experiments may not have definitively ruled them out. Finally, I will discuss about the generalizations to multi-layer states as well as the duality between the interlayer pairing and interlayer tunneling problems.