How can we establish the trust?
People often trust something blindly because they are told by some authority or elites. However, the authority or elites might cheat us for their own interest. Also, "we trust something because we are told to trust" is anything but a decent answer of this question. So the real answer is as below.
The answer turns out to be purely physical. Without an authority to command our thoughts, it can be proved that the only way to establish the trust is to sacrifice the real resource and show this sacrifice, a.k.a. proof-of-work. Meaning that, let everyone understand that to fake a trust is economically infeasible. Many ancient tribes had the same solution too: the way to allow someone to win others' trust is to show the public that he/she gives away his/her real resource, maybe his/her son or blood or time or something precious.
In bitcoin, here the real source to sacrifice is the electricity and time. To build a new block on top of the blockchain, it roughly takes 10 minutes of electricity of possibly more than the sum of top 500 super computers consuming. If you plan to fake a trust, you need to command more than half of the power just mentioned or set up a super computer with power more than the whole existing network. If you think this is costly and impossible for anyone on Earth, then you need to come to the conclusion that you trust the information stored in the blockchain to the degree of the correspondent cost. Blockchain technology is not green at all, however, trust is not "cheap" at the first place neither.
This is also the reason why I endorse bitcoin than many other cryptography currencies which follow the same blockchain technology used by bitcoin first. The sacrifice cost of bitcoin is the highest among them. If you don't trust bitcoin, you can not possibly trust other cryptography currencies. In other words, if you want to bet a success of one cryptography currency, bitcoin shall be the one.
This is also why I think blockchain application set up by some a few top banks is nonsense. Compared with bitcoin network, their computing power is so small and their members are so few that the sacrifice cost is so low and the risk of a cheated blockchain for their settlement application is no less than the risk of a cheated centralized settlement company fund by these banks which is cheaper and greener to setup.
This is also why to lower the sacrifice cost is not a way to scale up the adoption of bitcoin, as suggested by some people in the cryptography currency eco; the infamous Block Size War. We would have a bitcoin network that can compete with VISA in a much higher transactions capacity but then people would start losing the trust for the network because someone could fake the trust.
A bad corollary is that if someday we invent the "quantum computer" or anything that can do much more computation (in fact, 2^224 faster as explained here) in much less electricity, then that day would be the end day of bitcoin because the sacrifice cost is no longer economically infeasible. Of course at that time, we need some harder math than elliptic curve to build a new cryptography currency. However, if after all, all people and all governments understand and appreciate and follow the truth of money and interest rate and therefore bitcoin completes its mission, I would prefer not using a cryptography currency at all; a centralized database to keep our money number is much cheaper and greener if we trust it; "To trust or not to trust-that is the question".