Interleaved Mining

Proof of Work
Pros:

Battle tested with largest market cap coin, is secure and works
Requires an expense of resources external to the coin economy, ensuring that miners can only choose to extend one path on the chain at any point of time
With the growing role of decentralised electricity generation via Solar/Wind, has potential to be more decentralised in future
Miners have serious skin in the game (huge capital investment in hardware, cooling, location, etc..) with potentially low margins. It is in their interest to think long term to be able to recoup their investment.

Cons:

Extremely wasteful compared to a centralised solution (aka SQL database)
Environmental concern with energy burned
Advent of Pooled mining has caused excessive centralisation
Large solo miners further threaten decentralisation
Maturation of industry has resulted in relatively large barriers to entry (mining hardware purchase, sourcing of cheap electricity)

Proof of Stake
Pros:

Far cheaper than POW
Low barrier to entry - Allows anyone with coins and running a fullnode to participate
Pooled mining not possible without custodial risk?
More decentralised because it incentivises currency hodlers to run a full node for mining rewards

Cons:

Insecure - All solutions thus far require some form of centralisation (authority/checkpoints) or trust, something which is unacceptable for Bitcoin.
Nothing at stake problem allows large stakeholder miners to mine all possible forks, eventually find a fork where they come out on top, allowing them to rewrite history at point in future.
Not suitable for coin distribution as the first recipient of coins can always rewrite the entire history at any point in the future due to the fact that he owned 100% stake at one point of time.
Unlike POW, cannot objectively determine best/longest chain as there is no real cost to produce a larger chain(Nothing at stake). Resulting in checkpointing by trusted parties

Interleaved mining
By combining both mining methods, we can resolve or diminish many of the cons of both methods, while retaining all of their benefits.

The idea is simple, to alternate the mining algorithm between POW and POS every block (ie POW for all odd blocks and POS for all even blocks). POW blocks will retain all inflation subsidies to uphold the social contract promised by the original protocol. Transaction fees will be retained by the respective miners of the blocks, as is the case today. Since POS blocks will not receive new coins, the POW block reward will be doubled to stick to the original coin distribution schedule.

Block intervals will remain at 10 minutes.

To resolve block propagation delays causing POW miner centralisation, POS blocks will have a smaller block size limit to allow smaller miners to compete more equitably. This block size should be low to ensure that smaller POW miners are able to download and validate the block as quickly as possible such that orphan risk is extremely low. On the other hand, POW blocks may be considerably larger as POS miners are much less affected by propagation delays.

http://i.imgur.com/MHV490q.png

The addition of POS blocks in between POW blocks has many advantages, chiefly that having 51% of mining hashpower is no longer sufficient to control the network. For example, even if a miner had 51% of hashing power, he will not have 51% of stake(combination of coin age and coin quantity) to produce a valid POS block, allowing the rest of the network to produce a valid POS block and continue building atop the chain.

The key advantage is that by adding another dimension to mining, we make it much harder for anyone to control 51% of block generation beyond a single block.

Introducing POS blocks will greatly incentivise the running of full nodes and increase the decentralisation of validation of the chain.

In pure POS mining, it is possible for a POS miner to in future come up with a longer valid chain, however with interleaved mining, this is not practically possible as the next POW block will quickly seal the chain with proof of work, thereby rendering the attack unviable.

Motivation
The goal of introducing POS blocks is not for POSs sake, rather it is to increase decentralisation of mining and fullnodes by allowing smaller alternate blocks (POS blocks) to incentivise running of full nodes which at the same time allow rapid propagation of the blocks to POW miners, thereby preventing implicit selfish mining attacks caused by propagation delays.

Benefits

Solves 51% attack problem by adding another dimension of miners (POS)
Incentivises full nodes via POS mining, thus increasing decentralisation
Maintains Status Quo for POW miners by keeping inflation subsidy rewards solely for POW miners
Halves the running cost for POW miners while keeping revenue almost the same, potentially doubling profits for POW miners in the short term.
Hardens the Bitcoin blockchain by adding 2 dimensions of mining
Nothing at stake problem is solved by interleaving POS blocks with POW blocks, preventing a rewrite of history by any large holder. Any attack on POS or POW for that matter, only has a 1 block window, before the alternate mining method takes over.
POS Grinding attack is not possible as POW blocks serve as objective checkpoints
Solves large block propagation problem by keeping POS blocks smaller to ensure fast propagation to POW miners, allowing smaller miners to compete more equitably, thereby allowing larger blocks

Cons
Requires a hardfork