This paper introduces BlockReduce, a Proof-of-Work based blockchain system which achieves high transaction throughput by means of a hierarchy of merged mined blockchains, each operating in parallel on a partition of the overall application state. Most notably, the full PoW available within the network is applied to all blockchains in BlockReduce, and cross-blockchain state transitions are enabled seamlessly within the core protocol. This paper shows that, given a hierarchy of blockchains and its associated security model, BlockReduce scales superlinearly in transaction throughput with the number of blockchains operated by the protocol.