We desire the following primary goals:
Efficiency via proof of stake - the consensus should be secured without mining, thereby greatly reducing electricity waste as well as the need for large and indefinite ongoing ETH issuance.
Fast block time - the block time should be maximally fast, though without compromising security.
Economic finality - once a block is made, after a certain amount of time a state of affairs should arise where the bulk of the validators have "fully committed" to that block, meaning that they lose their entire ETH deposits (think: 10 million ETH of value) in all histories that do not have that block. This is desirable because it means that even majority collusions cannot conduct medium or long-range 51% attacks without destroying all of their ether; the default validator strategies are designed to be conservative about their willingness to make high-value commitments so the risk for honest validators should be very low.
Scalability - it should be possible to run the blockchain with literally no full nodes, ie. in a situation where all nodes, including validators, keep up with only a small fraction of the data in the blockchain and use light-client techniques in order to access the rest. This way the blockchain can achieve transaction throughput much higher than that of a single machine while at the same time ensuring that the platform can be run off of nothing more than a sufficiently large number of consumer laptops, thereby preserving decentralization.
Cross-shard communication - it should be maximally feasible to interoperate between applications that are on different parts of the state that are stored by different nodes, and to build applications that exist across multiple such portions of the state if an individual application's usage reaches such a point that a single node's computing power and bandwidth can no longer sustain it.
Computational censorship resistance - the protocol should be resistant to attempts by even majority colluding validators across all shards to prevent undesired transactions from getting into the chain and being finalized. This exists to some extent in Ethereum 1.0 via "censorship resistance by halting problem", but we can make this mechanism much stronger by introducing a notion of guaranteed scheduling and guaranteed cross-shard messages.