Proportional quota weighted voting system hierarchies

We investigate achievable and unachievable hierarchies in weighted voting systems where the quota is a proportion of the total weight. This extends work of Bean et al. (Theory Dec 65:285–302, 2007) where simple majority achievable hierarchies were characterized. In particular we resolve the question of which hierarchies are possible with common quota rules other than simple majority, such as 2/3-majority or 3/4-majority, and offer a conjecture about proportional quota achievable hierarchies in general.