Is "Causality" the equivalent of a claim that the future is predictable based on the present and the past? 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.
 A: 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.
A: You have to distinguish two different meanings of "causality". First, a theory is (Einstein-) causal if there is a light cone structure that connects events and all the possible influence of an event comes from the past light cone. 
Secondly, a theory can be called causal if for every single event we can identify a cause, or causal relationship. That means such events cannot happen spontaneously without any prior state leading to them. 
And then there's a third but wrong meaning of causality, that says cause comes before effect. This is meaningless because cause and effect are just defined by their very time order.(There's also a correct statement that goes in the same direction, namely saying that all observers agree on the order of cause and effect. But that's not important for the discussion at hands)
Another useful and related term is "determinism", which means that the future states of the system are a consequence of the current state only, and nothing more. A deterministic system does not have to allow the reconstruction of past states from the current state however, as it may lose information during the time evolution. To guarantee that the past is also 'predictable', you have to have a theory that is deterministic and its time reversal is deterministic too.
Now for general relativity you have both meanings of causality and determinism. That also implies that you can use a space-like time slice to describe the state of GR and predict all events from it. A past light-cone only works for the events contained in it.
Quantum theory is causal in the first case, meaning that events are only influenced by their past light cone. This is even true in the presence of nonlocal entanglement, because the actual interactions and therefore the state evolution are Einstein-local in relativistic quantum theory. For the second meaning of causality quantum theory does not seem to fit. Decay events seem to be uncaused and happen spontaneously. Quantum theory also appears to describe an indeterministic world, because we cannot predict the outcome of measurements and many events seem to be fundamentally random. 
The interesting question is, if these properties are just results of the incomplete knowledge of the observer about the state of the universe. Could it be possible that decays are really caused by some trigger that is too faint to be observed? Or would knowing the state of the universe allow to predict the outcome of a measurement as seen by an observer?
Many physicists argue that a fundamental scientific theory should be causal (in the 2nd sense) and deterministic, so there's a lot of research going on to answer these questions. The most important approaches are Bohmian mechanics which adds additional structure in order to get determinism, and the relative state or many worlds interpretation, which assumes a deterministic evolution of the universe and attempts to derive the observed indeterminism from it.
You can see that it is important to be very precise with the meanings of causality and determinism here. And in general one can say that the causality first meaning and second meaning together imply something even stronger than determinism, because they add the light cone structure. Determinism on the other hand does imply the second meaning of causality, but not the first, because a light cone structure is not necessary for determinism.
If you are interested in seeing how quantum theory can be thought of as a causal and deterministic theory, I would be pleased to welcome you to my blog at http://aquantumoftheory.wordpress.com
A: Causality (the effect occurs after the cause) is often used with the Kramers Koenig relations: http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations
Which in turns are often used in electromagnetism. As you can see, the structure of many usual functions is directly linked to the universe being causal.
From a purely sci-fi point of view, if at present, we only know the probability of a particle to be here or there, we would be foolish to ask where is the particle in ten years. The wiser question would be what is the probability of the particle to be here or there in ten years.
By the way, QM is causal, once you know the initial conditions you can compute whatever you want. 
A: Unlike quantum mechanics and relativity, causality is something we have a strong intuition for but somewhat weak theoretical basis, so I doubt you will get an answer that isn't a bit vague and philosophical. The problem is that the laws of physics are (mostly) time-symmetric, meaning that both past and future should be equally "predictable". That includes quantum mechanics if you don't believe in wave function collapse, and even general relativity (a time-reversed black hole is a white hole, which is not forbidden by GR). So what distinguishes cause and effect? The apparent distinction between past and future is called the arrow of time, which seems to be related to the second law of thermodynamics - the statistical tendency of systems to evolve from a more ordered state to a more disordered one. Why was the universe more ordered in the direction of time that we call the past? I don't think anybody knows.
A: An idealized computer is the prototypical example of a causal system. If you analyze the flow of information in a (potentially parallelized) computer program, it should basically be a directed acyclic graph. One way to destroy this property (and hence causality) for a computer program would be to run it in an endless loop and look for fixed points and other properties of this "iteration" instead of properties which depend on the actual number of the current time-step.
I expect from causality that (a generalization of) a directed acyclic graph captures the essence of the flow of information. A directed graph can be topologically sorted if and only if it is a directed acyclic graph. The ordering provided by a topological ordering can take over the role of time. A topological ordering is not unique, but this is no problem. Already special relativity tells us that time is not absolute. A slightly bigger issue is that there is a physical time, and it would be nice if it would actually provide a topological ordering, but in actual reality it probably just provides an approximate topological ordering.
I also want to clarify that determinism is no requirement for causality to me. It is fine if there is randomness, as long as the flow of information itself is "acyclic".
A: One of the lessons from SR is that if a causal influence propagated faster than $c$, observers would disagree on what occurred first, "cause" or "effect", i.e. cause and effect would be relative.
So, it might be argued that the principle of causality is perhaps equivalent to the statement that cause and effect are not relative.
I think it might be the case that the principle of causality is necessary but not sufficient to predict the future.
I, for one, believe that all actions are caused by entities but that the future is not determined. 
A: you are confusing determinisn with predictability, there are things that are deterministic (i.e. causal) but are unpredictable.
unpredictability does not imply the lack of a cause.
A: Cum hoc ergo propter hoc, or in English with it therefore because of it. Future might be correlated with the present and the past but not caused by it. So one could imagine a universe in which both the future and the past are caused for example by the state of the universe at some fixed point in time $t_0$. This could cause the universe to behave in a way such that for any two times $t_2>t_1$ the state of the universe at $t_2$ is in some way correlated with the state of the universe at $t_1$. Correlation is enough for prediction. In probably the most extreme case (and assuming an infinitely lived universe), we might even have $t_0 = \infty$, meaning the point in time that causes the correlation is infinitely far away. In other words, future can always cause the past, but still be inferrable from the past. So the answer to your question is no, predictability does not imply temporally ordered causality. The converse is true though, causality implies some degree of predictability. 
