Concurrency and scalability are fundamental properties of most complex systems. As a widely used formalism for modeling concurrency, Petri nets have been applied across various fields. Their mathematical abstraction, the vector addition system (VAS), has become a central object of study in theoretical computer science. The reachability problem of VAS, along with its algorithmic and complexity characterizations, has been regarded as one of the most fundamental and long-standing challenges in the field over the last 50 years. This study presents a comprehensive survey of the research on the complexity lower bounds of the VAS reachability problem. The definitions of VAS, their equivalent models, and the core verification problem, the reachability problem, are introduced. Known completeness results concerning the complexity of the reachability problem are reviewed. For the fixed-dimension case, proof frameworks based on multiplication triples and amplifiers are outlined, and the state-of-the-art achievements as well as core proof techniques are summarized. Finally, the current research bottlenecks, major open problems, and future challenges are discussed.