Curvature is a complex terrain derivative to compute, the equation that you use depends on the resolution of your input data, as you have to ensure that the curvature results you compute can be distinguished from noise in the data.
A lot of research has been done recently on curvature calculations on high resolution LiDAR data which showed that a scaling break exists at around 2 or 3 metres resolution and above this point more different algorithms (which I am not as familiar with) need to be used. The best information about calculating topographic curvature probably comes from Hurst et al 2012 and the references therein.
The basic principle of curvature calculation, as with slope and aspect, is to pass a moving window over the elevation surface and fit the elevation values to a 6 term polynomial function, the coefficients of which will yield the slope, aspect and curvature of the centre cell of the moving window.
ArcGIS uses a 3x3 search window which will only yield good results in areas completely devoid of vegetation, which makes the tool fairly useless unless people are aware of this limitation, this may suggest why it is not present in QGIS.
The maths was derived originally (I think) in Evans (1980) and was simplified in a few pages in Principles of Geographical Information Systems (Amazon link) which I can recommend as a good guide to this kind of terrain analysis at a basic level.
One way to calculate curvature of a DEM is to convert the DEM into an ascii raster, read it into a numpy array and then perform the polynomial fitting on a moving window passing through the data. This is fairly easy to do, but is very slow to execute and needs a fair amount of optimization (these kind of operations often get ported to c++ to speed them up).
To perform the operation in QGIS you can use the GRASS plugin r.slope.aspect which is also limited by the 3x3 fixed window.
I realize this is not the simple answer you were doubtless hoping for, but I hope you understand that curvature is complex to derive in a meaningful way. All the best.
Evans, I. S. (1980), An integrated system of terrain analysis and slope mapping, Z. Geomorphol., 36, 274–295.