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In this paper we show that all nodes can be found optimally for almost allrandom Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-timequantum spatial search procedure. This works for both adjacency and Laplacianmatrices, though under different conditions. The first one requires$p=\omega(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log(n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergenceof eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty$ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property doesnot hold for any matrix, due to the connectivity issues. Hence, our derivationconcerning Laplacian matrix is tight.

Vertices cannot be hidden from quantum spatial search for almost all random graphs