Surprisingly, in two-dimensional systems quasi-particles may exist that are neither fermions nor bosons. When these particles are interchanged, their joint quantum mechanical wave function is predicted to pick up any phase in between 0 (as for bosons) and pi (as for fermions), hence the name anyons.Even more intriguing is the class of non-Abelian anyons where interchange of particles completely changes the ground state of the system. This phenomenon lays at the heart of a wealth of theoretical proposals for new types of quantum statistics and topological quantum computation that is robust against decoherence. While theory has well advanced, experimental realizations possess their own challenges and are seriously lacking behind.It is the objective of this proposal to experimentally realize a platform to detect and control non-Abelian anyons. We propose to combine the particle-hole symmetry of a superconductor with the spin-momentum locking at the surface of a topological insulator. Topological Josephson junctions are predicted to host Majorana type bound states at vortices. We propose to artificially create Josephson vortices at the junction of three phase-biased superconducting islands and to control and braid multiple Majorana states to prove their non-Abelian anyon character.In preliminary experiments we have shown, as one of the first groups in the world, to be able to induce superconductivity in a topological insulator by the proximity effect. This puts our group in a unique position to open up the field of topological Josephson physics. The great technological challenges of the present proposal lay in the development of topological insulator materials with higher surface mobility and their integration into Josephson electronic circuitry with multiple phase biased superconducting islands. For the phase biasing as well as the read-out of the Majorana states after braiding on-chip SQUID based current amplifiers will be developed.