We discuss the relationship between fractional quantum Hall (FQH) states at filling factor nu = p/(2mp + 1) and quantum spin chains. This series corresponds to the Jain series nu = p/(2mp + 1) with m = 1 where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin S = 1 chains with p spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as p is increased which is similar to S = p integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.