Tom,
The wavefunction is complex. It consists of two real functions, usually called the real function and the imaginary function. But the absolute square is only a single real function.
The issue is not the number of points, but the number of independent points, required to reconstruct a curve. "A line of any length" is determined entirely by just its two end-points. But other curves require more points to specify and thus more information. Information has nothing to do with time. It is purely a mathematical concept: the specification of the number of independent points and the number of independent, significant bits per point, required to perfectly reconstruct an arbitrary curve, in which the definition of what is meant by "perfect" is very specific and rather unusual. It means if you recover the information content of both the original curve and the reconstructed curve, they will be identical, with no bit errors anywhere. In other words, a copy of a copy of a copy... is just as good as the original. What is unusual about this, is that the copy and the original need not appear to be identical at all, as one might naively expect. Noise and distortion may cause them to be non-identical, continuous functions. But the digital bit-streams extracted from them, by a process that knows the correct extraction procedure (think of error detection and correction) will nevertheless be identical in spite of their differences in appearance.
This casts what it means to be "identical" particles, in a rather different light, than that which is familiar to the physics community.
Rob McEachern