Is it possible to combine PoW with PoS by taking turns? I have read about cryptocurrencies and am very excited by many. I have read that one of the worries and constant debates is Proof of Work vs Proof of State with respect to securing the validity of the blockchain, especially within Ethereum.

I wondered why we would not just combine the two. For example, the first block validated by PoW, second block by PoS, third by PoW, etc. As each block depends on the previous a 51% attack would be a lot more difficult and power consumption significantly reduced. 

Is there an obvious problem with doing that?
Yes. The problem with this approach is that all PoW nodes will be idle half of the time, and all PoS nodes will be idle half of the time. This isn't a huge problem. You could mine a different coin during the time that your PoW miner was idle; PoS nodes being idle half the time is not a big issue.

Your proposal does have a security advantage that I don't think you've pointed out. Specifically, in order to conduct a 51% attack, you need to be able to produce both PoW and PoS. You can have 51% of each, or you can compensate for a smaller amount of PoW or PoS by having more of the other. 

For a 51% attack, ordinary PoW requires:


  W = Your PoW capacity, where 0.5 means you control half of the hashpower
  W > 0.5


For a 51% attack, ordinary PoS requires:


  S = Your PoS capacity, where 0.5 means you control half of the coins actively participating in PoS
  S > 0.5


For a 51% attack on this scheme, you'd need:


  1/((1/S + 1/W)/2) > 0.5