  We introduce two a posteriori error estimators for N\'ed\'elec finite element
discretizations of the curl-curl problem. These estimators pertain to a new
Prager-Synge identity and an associated equilibration procedure. They are
reliable and efficient, and the error estimates are polynomial-degree-robust.
In addition, when the domain is convex, the reliability constants are fully
computable. The proposed error estimators are also cheap and easy to implement,
as they are computed by solving divergence-constrained minimization problems
over edge patches. Numerical examples highlight our key findings, and show that
both estimators are suited to drive adaptive refinement algorithms. Besides,
these examples seem to indicate that guaranteed upper bounds can be achieved
even in non-convex domains.
