The vast world of quantum advantage

The quest to identify quantum advantages, where quantum physics truly outperforms classical physics, lies at the heart of quantum technology. While quantum devices promise extraordinary capabilities, from exponential computational speedups to unprecedented measurement precision, distinguishing genuine advantages from mere illusions remains a formidable challenge. In this endeavor, quantum theorists are like prophets trying to foretell a future where quantum technologies reign supreme. Yet, the boundary between visionary insight and unfounded fantasy is perilously thin. In this perspective, we explore the properties defining an ideal quantum advantage and examine our mathematical tools for navigating the vast world of quantum advantages across computation, learning, sensing, communication, and beyond. We show that some quantum advantages are inherently unpredictable using classical resources alone, suggesting a landscape far richer than what we can currently foresee. While mathematical rigor remains our indispensable guide in this exploration, the ultimate power of quantum technologies may emerge from the quantum advantages we cannot yet conceive.