  We discuss the relationship between fractional quantum Hall (FQH) states at
filling factor $\nu=p/(2p+1)$ 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.
