When reading about dialectics, one repeatedly hears that change and contradiction are intimately and necessarily connected, even for the simplest forms of change.
For example, Engels writes that "Motion itself is a contradiction: even simple mechanical change of position can only come about through a body being at one and the same moment of time both... in one and the same place and also not in it." I don't want to examine this particular physical case too closely, because it is entangled with quantum mechanics (via the quantization of space and time), and this is a whole question in itself. But Engels' statement is nevertheless interesting.
Engels' claim relies on the obviousness of one kind of "contradiction" - that which results from change. A body is in one place; later it is not in that place but in another place. Thus a statement which was once true is no longer true, and we have two true but incompatible statements - a "material contradiction", perhaps? Trotsky finds the same kind of "contradiction" in a pound of sugar: "All bodies change uninterruptedly in size, weight, colour, etc. They are never equal to themselves."
But this kind of contradiction is trivial and uninteresting. Any change involves an entity or process moving from one state to a different state. We have already rejected the idea that literally everything changes, and Trotsky concedes that even when things do change, if the changes are "negligible for the task at hand", we can ignore them, and speak of coherent, "self-identical" entities. So if this is all there is to the relationship between contradiction and change, talk of "contradiction" adds nothing, and we are left with nothing more than an admonition not to assume that things are static when it is not safe to do so.
Engels' non-trivial claim is that change "can only come about through" a contradiction which already exists, embedded in reality at a given instant. Bertell Ollman puts the same point clearly: "Dialectical thinkers attribute the main responsibility for all change to the inner contradictions of the system or systems in which it occurs." This claim, that change necessarily results from contradiction, is the one that seems like it might have really interesting implications.
To evaluate it, we first need to define "contradiction", or material contradiction specifically, more precisely.
One approach would be to start with the relatively clear definition of a logical contradiction - "a proposition, statement, or phrase that asserts or implies both the truth and falsity of something" - and say that a material contradiction is simply a logical contradiction which is true. This is not promising, for two reasons.
The first is that for any of the examples we have used so far, one can disentangle them so that there is no actual logical contradiction. "Capitalism has both an inherent drive to expand and an inherent tendency to crisis which restricts expansion." "Capitalism both empowers its bourgeois rulers and, ultimately, leaves them helpless before their natural enemies, the working class." We are not actually asserting simultaneously the truth and falsity of any proposition in these rhetorical oppositions - we are merely looking at different aspects of the situation.
The second reason not to claim that logical contradictions can be true is the principle of explosion: from a true contradiction, one can derive anything. (And not in a useful way, contra XKCD.)
For example, say that a pound of sugar is itself, and it is not itself. Then it is itself - that is one half of what we have just asserted. Then either the pound of sugar is itself, or Santa Claus exists - to any true statement, one can add "or [whatever]", and the whole, the disjunction, will remain true. But then let us remember the second half of the original contradiction: the pound of sugar is not itself. With this, we can reject the first half of our disjunction. And since we have shown that the disjunction as a whole is true, that means the second half must be true. In other words, we have just proven that Santa Claus exists.
Suspect a trick? Of course, this reasoning isn't valid. But that's because the premise wasn't valid. Each step thereafter used a rule of propositional logic which we rely on all the time and would be very difficult to do without.
So let's abandon the idea of true logical contradictions and look for another definition of a material contradiction.
We could try to retreat a little, without changing direction, and say that material contradictions occur whenever two statements which are somehow contrary or in tension, without being strictly contradictory, are both true. This, however, is almost as bad an option, because there is no good way to define this kind of propositional tension - it will likely come down in each case to rhetoric, whether English offers a way to phrase the two statements so that they sound contradictory.
A more promising approach, I think, is to view a material contradiction as the existence of opposing forces or tendencies. Ollman gives a good definition of a contradiction along these lines: "The incompatible development of different elements within the same relation." Ollman's phrase "incompatible development" is nice because on the one hand we can use the strict sense of "contradiction" - the outcomes towards which various forces tend are genuinely incompatible, that is, they could not occur simultaneously without logical contradiction - and on the other hand we do not have to claim that a logical contradiction is ever actually true, but rather that contradictory forces must result in a change in the system before the state of logical contradiction is reached.
Let us provisionally accept this definition, then, and go back to our original question: does all change result from this kind of contradiction?
Our definition of contradiction excludes one semi-intuitive rationale for the "change requires contradiction" thesis - the notion that change requires some sort of "lack", or imperfection, in what exists. The idea is that a truly complete and self-consistent reality would necessarily be static - from perfection can flow only constant perfection. But given the idea of contradiction as the existence of incompatible tendencies, there is no need to try to explore and clarify this rationale - a "lack" is not by itself a contradiction.
Losing one possible rationale, however, is not necessarily a problem for the original thesis. So let's consider what it implies.
It is at least plausible that all change requires some sort of pre-existing tendency, or else it would be uncaused. The only counter-example I can think of is if there is true randomness, as in some interpretations of quantum mechanics, but even there we might describe the probability distribution which governs the quantum state as an existing tendency, and so preserve our claim.
It is also plausible that all change occurs within some system. Now, certainly all change within any given system does not necessarily stem from a cause internal to that particular system. If the geologists are right, Earth's ecology in the era of the dinosaurs was transformed from the outside in about as dramatic a way as might be imagined - by the impact of a gigantic meteor. And there are many other similar examples, phenomena for which I like the term "Excession", following Iain Banks. But there is always a larger, enveloping system, to which any given cause of change is internal, up to the scale of the universe.
In any case, that still leaves one more step before we have a positive answer to our question - does all change, that is caused within a system, stem from multiple, contradictory tendencies or developments? Or, equivalently, since we have admitted a necessary connection between tendency and change - is every tendency in reality matched by one or more opposing tendencies within the same process?
I think not. Take Engels' simple object in motion - for it to travel in a straight line, there need be no tendency other than its inertia. Or at a more abstract and social level, take the tendency for scientific knowledge and technology to advance. Certainly that does not have only positive effects, or even necessarily advance the average person's knowledge. But is there a counter-tendency for technology to regress?
For any example I can name, I am sure someone can come up with some contrary tendency somewhere. But remember that we are not merely seeking a contrary tendency, we are seeking one within the same system, the same process of change. That, I do not think it is always possible to provide.
One more example: a computer running a sort algorithm on a list of numbers in its memory, say merge sort. There is a tendency, as this process runs, for the list to become sorted - a tendency which is in fact mathematically provable, and can be quantified in various ways (for example, we can find the number of comparisons which must be done in the worst case for a list of any given length). Here we have a well-defined process - in fact, "process" could be a technical term here as well as a philosophical one - which not only contains a well-defined development, but apparently excludes any possibility of a counter-development.
Remember also the dilemma posed in the previous post - in relating dialectics to science, we want to avoid either Lysenkoism or mere mysticism. Now that we have made the question of whether change always stems from contradiction more concrete, we can see that the same dilemma applies here. Either dialectics commands scientists to find opposing tendencies in every system, or it asserts their existence without allowing any concrete conclusions to be derived from this assertion. Neither option is satisfactory.
At best, then, if a conception of dialectics as a set of laws or facts about the world is defensible, its defense requires losing content. We cannot sustain at the same time two universal propositions; that change is always caused by tendencies which are 1) united and internal to a single process, and 2) contradictory. To make either proposition universal we must abandon the other, and doing so would leave dialectics with little to say.
Can this problem be solved by narrowing dialectics' domain? Perhaps we can retain the idea of dialectics as a set of truths about the world, if we speak only of a part of the world - human history, or capitalism. Or do we have to abandon dialectics' claim to describe reality, in favor of a conception of dialectics as method, or form, or critique? These questions still have to be answered.
 Anti-Duhring, chapter 10.
 One can certainly view something like Schroedinger's cat as a case of a material contradiction. But I think it is more useful to view it as an indication that we should not try to think of quantum mechanical particle as analogous to macro-scale objects like cats.
 "The ABC of Materialist Dialectics", in A Petty Bourgeois Opposition in the Socialist Workers' Party.
 Dance of the Dialectic, p. 18.
 Merriam-Webster's Online Dictionary. (Yeah.)
 A few mathematicians have tried to develop logics which allow contradiction while avoiding explosion, most commonly logics which take more values than "true" and "false". But if consistent, they tend to end up adding complexity without having any actual advantages in terms of conceptual power. Timothy Williamson's book Vagueness has a persuasive section on this.
 Dance of the Dialectic, p. 17.
 Hegel uses the term "deficiency": "Internal self-movement proper... is nothing else but the fact that something is, in one and the same respect, self-contained and deficient, the negative of itself." Cited in Rees, Algebra of Revolution, p. 51.