Despite the tremendous interest in cryptocurrencies like Bitcoin and Ethereum today, many aspects of the underlying consensus protocols are poorly understood. Therefore, the search for protocols that improve either throughput or security (or both) continues. Bitcoin always selects the longest chain (i.e., the one with most work). Forks may occur when two miners extend the same block simultaneously, and the frequency of forks depends on how fast blocks are propagated in the network. In the GHOST protocol, used by Ethereum, all blocks involved in the fork contribute to the security. However, the greedy chain selection rule of GHOST does not consider the full information available in the block tree, which has led to some concerns about its security. This paper introduces a new family of protocols, called Medium, which takes the structure of the whole block tree into account, by weighting blocks differently according to their depths. Bitcoin and GHOST result as special cases. This protocol leads to new insights about the security of Bitcoin and GHOST and paves the way for developing network- and application-specific protocols, in which the influence of forks on the chain-selection process can be controlled. It is shown that almost all protocols in this family achieve strictly greater throughput than Bitcoin (at the same security level) and resist attacks that can be mounted against GHOST.