How does this compare to Avax/Avalanche concensus protocol ? 
Great question. I appreciate the work Ava labs has been doing. Back when he was still at Cornell, Dr. Sirer provided some valuable guidance in early stages of my research, notably introducing Snow White Proof-of-Stake by Rafael Pass and Elaine Shi. The two protocols can be distinguished primarily on the basis of their security model: (Probabilistic finality vs Blockchain finality)
Ava employs random sampling of a BlockDAG in order to determine what transactions were seen first. Crystal, however, ascertains linear order for the DAG by using Proof-of-Work. Both protocols utilize Proof-of-Stake to compile transactions into blocks, but take distinct approaches to compose these blocks into a cohesive chain.
AVA relies upon the assumption that a super-majority (2/3) of validators are honest, whereas Crystal only requires an honest majority (51%) to achieve trust in the network. Ava's security therefore hinges heavily on the ongoing integrity of the validator pool, with few mechanisms to recover the network in the event of an attack; this is partly why the AVA foundation retains control of a large portion of protocol tokens.
On the other hand, Crystal could be deployed on an existing system, like bitcoin, without a disproportionate pre-mine. By leveraging the hybrid PoW/PoS approach, our proposed system is also capable of horizontal scaling (or sharding), something which I believe they are still trying to figure out. In summary, AVA works around the notion of 'sub-nets' for scalability, but due to limitations of their present design, this data is not consensus-critical; sub-nets can be manipulated independently from the rest of the network. Crystal's shards do not have this issue, as sub-chains remain involved in global blockchain consensus.
Some of these ideas are borrowed from Hybrid Consensus (also by Pass and Shi), which is summarized in the Building Blocks section of the Crystal whitepaper. Under 2/3 honest majority, both networks can achieve optimal responsiveness ('instant' tx finality). However, our design features a mechanism to sustain this performance securely across an expansive, fast-growing network using Bitcoin's original security model.
