Abstract
<jats:p>Quantum technologies are moving from foundational science toward engineered systems in sensing and metrology, computing and simulation, communication and networking, and security. The transition is uneven in technological maturity: atomic clocks and several quantum sensors are operational; quantum key distribution has reached field deployment but remains constrained by integration, assurance and network architecture; post-quantum cryptography is entering standards-based migration; and general-purpose fault-tolerant quantum computing remains an engineering objective despite major progress in error correction. This treatise develops a mathematically grounded and application-oriented account of what quantum technologies are, what physical resources they exploit, what advantages have been proved or demonstrated, and where claims remain speculative. It distinguishes among scientific feasibility, engineering readiness, operational utility, economic value and societal legitimacy. Across fundamental physics, Earth and space sciences, environmental monitoring, health and life sciences, cybersecurity, critical infrastructure, hazards, defense, finance, decision support and AI-enabled systems, this work identifies bottlenecks and proposes a systems roadmap for robust, scalable, efficient, verifiable and responsible quantum-enabled capabilities.</jats:p>