Superconductors and superfluids have ever since their discovery been of great academic interest in physics. This is reflected by the fact that twenty people have been awarded Nobel prizes for studying such systems during the last century, yet the most outstanding problems in the field remain open. The 2016 Nobel Prize in physics was given for the first works on topological phase transitions in superfluids and topological states of matter. Such phase transitions are driven by the proliferation of topological defects or vortices. In recent years many novel superconducting materials were experimentally discovered where the topological defects are much more complex and not understood. The current proposal (i) confronts outstanding open questions in superconductivity and (ii) focuses on predicting a new type of superconducting and superfluid behavior. In recent years, there were many experimental breakthroughs in condensed-matter systems that exhibit multicomponent many-body degrees of freedom, making it a rich emergent field of research. As a consequence of having several types of carriers responsive for superconductivity, these materials behave fundamentally differently from traditional superconductors. These superconducting states require theoretical understanding from the fundamental viewpoint, and also in order to use such materials in practical applications, which we plan to address in the project. The proposal's third focus area is the recent prediction that superconductors are the best topological materials that can be used to build qubits for quantum computation. This started the pursuit of using these states to implement qubits for quantum computers and quantum emulators, which forms a growing part of this project.