**Remote blockchain.**InRemote blockchain. In the current implementation of the mainline client, every client downloads and store a complete copy of the blockchain. This is the best way to boostsrap the bitcoin network while transaction volume is low. However at peak transaction volume most users will have no interest in downloading and storing TBterrabytes worth of data. In the future if a client trusts an online entity they could query transactions against that cliententity rahter than store the block chain locally. Obivously this requires a level of trust so it would be important to choose a trustworth "block chain provider" however it has the advantage of each client only needing to query transactions they are interested in (for example client could ask for all transactions involving a particular address (to get current balance) or number of confirms for a particular transactions (to update the confirmation status). Before someone cries "centralization" there would be no official blockchain provider but instead numerous independent entities offering this as a service (either paid or free). If someone is paranoid they could use/build a client which queries two different block chain providers to ensure they return same data. Lastly if a person is sufficiently concerned about security they could trust no-one and compute the entire blockchain themselves.
GPU Acceleration of transaction verification. GPU are currently used to find a hash to sign the block as this is cryptographically intensive and any speed upperformance increase pays huge dividends. This isn't true for transaction verification;verification as transaction volume is low. As a result the bitcoind uses non-openCL CPU calculations to verify transactions. There is no reason for faster verifications because today transaction volume is relatively low. GPU If necessary an Open-CL GPU powered client would be able to handle a magnitude more transactionsverify significantly higher transaction volume.
A rough estimation of future computation power required. The paper linked by author indicates it would take a modern CPU can only handle ~100 tps (transactions per second). For integer math (like used in cryptographic hashing) a high end GPU is (as of writing) roughly 20x as powerful as a high end CPU. That allows in the ballpark of ~2000 peak tps per GPU. Furthermore even with say 50% annual transaction volume growth we are two decades away from VISA level transaction volume. Moore's law would indicate that average CPU/GPU in 20 years would be roughly 1000x as powerful. That would be on the range of 100,000 tps per CPU and 2 mil tps per GPU in 20 years. Even if growth in computational power slows as long as it is not likely to beisn't slower than the rate of transaction volume so likely over the next two decades despite growinggrowth in transaction volume it will be possible for a single CPU (and eventually a single GPU) to verifyensure that all transactions can be verified by off the shelf hardware.
Actual transaction volumes. Bitcoin is digital cash. Many people today chose not to use cash for online and offline transactions favoring instead indirect payment methods (check, debit card, credit card, gift card, etc). It is probable that higher level payments systems will emerge (even possibly those based on credit worthiness) that workbuilt on top of bitcoin network will emerge. For users Users of these systems bitcoin would only be used for settlement of dailyshould cause significantly lower transaction volume (or monthly bills) between entities. This significantly reduces the amount of transactions necessary. As For an example the average userlets look at how VISA transactions work. An individual may have 500200 VISA transactions in a month but only makes a single payment to the card issuer (a 200:1 multiplier between purchases and actual currency movement). Likewise a company may have a thousands of VISA transactions in a day but recives that as a single currency payment (usually ACH). Another thing to consider is "VISA level" is more an end game scenario. A more likely interim goal (say over next 2 decades) would be a PayPal"PayPal level transaction volumevolume" which despite being a modest 100 tps (vs VISA >4K tps) would indicate Bitcoin is a massive success.