I've been trying to look for some evidence showing that PoS is more energy-efficient than PoW. In theory, it's true, but if there are any figures or stats you can share on this, I'd be very grateful.
The simplest way to explain this is to consider the energy required to produce 1 hash, and then look at the % of hashes created for each block in the block-chain, and the number of hashes that failed to meet the Target.
PoW mining hardware is typically much more efficient at hash creation, being based often on ASIC, but consider that it usually takes just one hash below Target to create any block. The Bitcoin hashrate is estimated around 100 Exahashes per second, or 100,000,000,000,000,000,000, and for a 10 minute block spacing multiply by 600 seconds, so its an understatement to say that a huge amount of hashes are "wasted".
Now consider a Proof of Stake scheme, based on Bitcoin (same number of coins, same block spacing). Its common for the minimum stake input size to be set to 0.01 coin, and there isnt much disadvantage to holding larger amounts. In PoS you can create as many hashes on your node as you want, but you can only create unique hashes for each UTXO that you have. Assuming all 21,000,000 coins are in circulation and all are in UTXOs exactly equal to 0.01, and everyone is staking, the maximum hashrate possible is 2,100,000,000/s. So PoS "wastes" a lot less hashes to reach the next block. Consider its unrealistic that 100% of holders would be willing to or able to stake, coins may be on exchanges, lost keys, technical limitations, small/large UTXOs etc.
ASICs are much more efficient than an ordinary x86 CPUs at creating hashes, but consider that most ordinary CPUs are idling 99% of their working lives, and an additional few hundred hashes per second is not adding any significant energy budget. You can run a PoS staking node on 1 core of a Raspberry Pi 2b, which has a maximum energy draw of 5 Watts and it will run at significantly below maximum energy draw, assuming you are using that Pi for other tasks, running a PoS node is practically energy free for a single user. But lets err on the side of caution and say a PoS staking setup, including router, hard drives, additional RAM (for longer chains) etc. requires 100Watts. If we assume at scale 25% of coins are non stake-able on exchanges, and a further 25% are in UTXOs too small to stake, and the average staker has 1 coin; that leaves around 10 million staking nodes * 100 Watts per node * 24 hours a day * 365 days a year = 8.7 TWh per year. Bitcoin is currently at 75 TWh per year and constantly rising.
In both cases security comes from the cost to attack the network, in PoW we have a situation where there is massive cheap surplus energy available for someone to be able to burn more hashes (solar energy is becoming cheaper than fossil fuels at scale), whereas for Proof of Stake the resource is finite, the supply of coins is hard limited in the protocol itself. If we believe in supply/demand creating price conditions, the cost to attack PoW long term can at best stabilize as new energy supply meets demand, but in PoS as the coin price increases, it gets more expensive to acquire additional coins (beyond theft) to be able to gain a significant enough stake to try and break consensus.
When providing an equivalent amount of security, I don't believe PoS can be more efficient than PoW. I do understand how it may appear this way on the surface, but I think deeper consideration shows that this is a fallacy.
The security of Bitcoin (PoW) is dependant on the economic incentives built into mining: it is the cost of attacking the system vs. the expected return that matters, not the total amount of energy spent on mining. Of course energy has a cost, but it is important to note that it is the cost, not the energy, that is important when considering the security of the network.
Therefore, we can define a system's security relative to the cost of attacking it successfully. Less cost to attack = less secure, and vice versa.
PoW mining is a permission-less activity, anyone is free to join, and so we expect to reach a free market equilibrium: the cost of mining $1 worth of Bitcoin is expected to trend towards $1. If there is profit to be made, we would expect more miners to join the market. If costs outweigh returns, we would expect miners to leave the market.
In PoS systems, we expect a similar result: participants will be willing to incur costs of up to ~$1, for every $1 worth of rewards they receive in return. You wouldn't want to pay someone $1 to provide only $0.50 worth of security, and likewise, validators won't want to get paid $1 to provide $2 worth of security.
In PoW systems, a very large portion of a miner's cost is the energy used to perform computational cycles. There are additional costs to consider (hardware, overhead, employees, maintenance, etc), but the most efficient miners will be expected to minimize these costs, and spend as large a portion of their budget on energy as possible (thus maximizing their expected return).
PoS seems to be more efficient, due to an apparently minimal number of computational cycles (compared to PoW), but when considering the above, we can arrive at a more nuanced understanding:
If a PoW network and a PoS network were measured to have the same amount of security (denominated in your favourite currency), we should expect each network's participants to incur costs that approach the reward to create and maintain this security.
With this in mind I think it is interesting to reflect back upon the question at hand: how do we actually calculate energy efficiency? Is the upfront cost of the electricity needed to power the computers that run the network really the whole story? Shouldn't we really just be striving to maximize the security provided per unit of energy spent? Keep in mind that energy can be spent in many different ways, it is not limited to just the electricity powering the computers involved. In terms of creating a more efficient network, that is what will ultimately matter.
Creating complex systems is energy intensive, and the laws of thermodynamics guarantee that a large amount of the energy used to create such a system will be lost as 'waste' (heat, etc). So using energy to very directly create security through PoW hashing will be much more efficient than using that energy to create the (complex) social constructions needed to enforce a PoS mechanism. This means that with PoW we can get more security for less energy, and thus have more energy left over for humanity to enjoy otherwise.
A further consideration: a large portion of a PoW miner's costs come from an external free market for energy, instead of an internal staking system. There is a cost to security, and so having as much of that cost tied to something external and universally accessible (ie, energy) as possible seems like the most logical decision. If it is more difficult to participate in the 'validation market' (ie, act as a miner/staker), then we will expect more opportunity for validators to make profit, and thus the security of the network will be lower. This sort of 'red tape friction' creates a less efficiently secured network, dollar for dollar.
As humanity has progressed there has been an undeniable trend of an increased capture and utilization of energy, and this is what allows us to advance technologically over time. And yet still, we utilize such an incredibly tiny fraction of the sun's output, that we are nowhere near being about to 'run out of energy'. The variable that is more obviously limited is our capital (which is created via the consumption of energy), and so we should strive to use the energy that we do capture as efficiently as possible, in order to maximize our capabilities as a technological species.