Local certification is a topic originating from distributed computing, where a prover tries to convince the vertices of a graph G that G satisfies some property  \(\mathcal {P}\) . To convince the vertices, the prover gives a small piece of information, called certificate, to each vertex, and the vertices then decide whether the property \(\mathcal {P}\) is satisfied by just looking at their certificate and the certificates of their neighbors. When studying a property  \(\mathcal {P}\) in the perspective of local certification, the aim is to find the optimal size of the certificates needed to certify  \(\mathcal {P}\) , which can be viewed as a measure of the local complexity of  \(\mathcal {P}\) . A certification scheme is considered to be efficient if the size of the certificates is polylogarithmic in the number of vertices. While there have been a number of meta-theorems providing efficient certification schemes for general graph classes, the proofs of the lower bounds on the size of the certificates are usually very problem-dependent. In this work, we introduce a notion of hardness reduction in local certification, and show that we can transfer a lower bound on the certificates for a property  \(\mathcal {P}\) to a lower bound for another property  \(\mathcal {P}'\) , via a (local) hardness reduction from  \(\mathcal {P}\) to \(\mathcal {P}'\) . We then give a number of applications in which we obtain polynomial lower bounds for many classical properties using such reductions.