[simias]

What do you mean about "cost of verification"? Verification is not too difficult, it's reaching a consensus among trust-less nodes that is.

PoW solves the problem by making it so that any node which receives two valid but conflicting versions of blockchain has an objective metric to decide which one is the "right" one. The answer being whichever has the most work put into it. Since you can't fake work you can't arbitrarily create a new chain that would take over the others (unless you manage to work harder than all the rest of the network combined, hence the 51% attacks).

Without PoW if you receive two valid but conflicting chains you need an other metric to decide which one you select. This paper describes such an approach.

[stri8ed]

Verification becomes difficult, when the system gains adoption and processes thousands of TX per second. Then the issue becomes about the trade-off between node resource requirements, and decentralization. Fundamentally, every node having to process/verify every single transaction in the network,is not scalable.

[SkyMarshal]

Hence the ongoing debate around whether verification should be less computationally costly than computation (BTC) or whether a system can be architected to successfully scale even when verification == computation (ETH).