In classical (Newtonian) mechanics, every observer had the same past and the same future and if you had perfect knowledge about the current state of all particles in the universe, you could (theoretically) compute the future state of all particles in the universe.
With special (and general) relativity, we have the relativity of simultaneity. Therefore the best we can do is to say that for an event happening right now for any particular observer, we can theoretically predict the event if we know everything about the past light cone of the observer. However, it tachyons (that always travel faster than the speed of light) are allowed, then we cannot predict the future since a tachyon can come in from the space-like region for the observer and can cause an event that cannot be predicted by the past light cone. That is, I believe, why tachyons are incompatible with causality in relativity. Basically, the future cannot be predicted for any given observer so the universe is in general unpredictable – i.e. physics is impossible.
Now in quantum mechanics, perfect predictability is impossible in principle. Instead all we can predict is the probability of events happening. However, Schrodinger's equation allows the future wavefunction to be calculated given the current wavefunction. However, the wavefunction only allows for the predictions of probabilities of events happening. Quantum mechanics claims that this is the calculations of probabilities is the best that can be done by any physical theory.
So the question is: "Is the predictability of the future to whatever extent is possible (based on the present and the past) equivalent to the principle of causality?" Since prediction is the goal of physics and science in general, causality is necessary for physics and science to be possible.
I am really not asking for a philosophical discussion, I want to know if there are any practical results of the principle of causality other than this predictability of the future of the universe. Please don't immediately close this as being a subjective question, let's see if anyone can come up with additional implications for causality besides future predictability.
Best Answer
Your question "Is the predictability of the future to whatever extent is possible (based on the present and the past) equivalent to the principle of causality?" has the trivial answer ''no'' as the qualification ''to whatever extent is possible'' turns your assumption into a tautology. The tautology makes your statement false, as your question asks whether the universally true statement is equivalent to causality. An answer "true" would make any theory causal, thus making the concept meaningless.
Why is your assumption a tautology? No matter which theory one considers, the future is always predictable to precisely the extent this is possible (based on whatever knowledge one has). In particular, this is the case even in a classical relativistic theory with tachyons or in theories where antimatter moves from the future to the past.
However, in orthodox quantum mechanics and quantum field theory, causality is related to prepareability, not to predictability.
On the quantum field theory level (from which all higher levels derive), causality means that arbitrary observable operators $A$ and $B$ constructed from the fields of the QFT at points in supports $X_A$ and $X_B$ in space-time commute whenever $X_A$ and $X_B$ are causally independent, i.e., if (x_A-x_B is spacelike for arbitrary $x_A\in X_A$ and . $x_B\in X_B$.
Loosely speaking, this is equivalent to the requirement that that, at least in principle, arbitrary observables can be independently prepared in causally independent regions.
Arguments from representation theory (almost completely presented in Volume 1 of the QFT books by Weinberg) then imply that all observable fields must realize causal unitary representations of the Poincare group, i.e., representations in which the spectrum of the momentum 4-vector is timelike or lightlike.
This excludes tachyon states. While the latter may occur as unobservable unrenormalized fields in QFTs with broken symmetry, the observable fields are causal even in this case.